Alternating Current

Alternating Current

The basic principle of the ac generator is a direct consequence of Faraday's law of induction. When a conducting loop is rotated in a magnetic field at constant angular frequency ω a sinusoidal voltage (emf) is induced in the loop. This instantaneous voltage is,

            V = V₀ sin ωt                                       ... (i)

The usual circuit diagram symbol for an ac source is shown in Figure.

In Equation (i) V₀ is the maximum output voltage of the ac generator, or the voltage amplitude and w is the angular frequency, equal to 2p times the frequency f.

            ω = 2pf

The frequency of ac in India is 50 Hz, i.e.,

            f = 50 Hz

So,       ω = 2pf » 314 rad/s

The time of one cycle is known as time period T, the number of cycles per second the frequency f.

            T = 1/f     or  T = 2π/ω

A sinusoidal current might be described as,

            i = i₀ sin ωt

If an alternating current is passed through an ordinary ammeter or voltmeter, it will record the mean value for the complete cycle, as the quantity to be measured varies with time. The average value of current for one cycle is,

Thus,      _{one cycle} = 0

Similarly, the average value of the voltage (or emf) for one cycle is zero.

              _{one cycle} = 0

Since, these averages for the whole cycle are zero, the dc instrument will indicate zero deflection. In ac, the average value of current is defined as its average taken over half the cycle. Hence,

This is sometimes simply written as, i_av. Hence,

               i_av = _{Half Cycle} = 2/π i₀ ≈ 0.637 i0

Similarly, V_av =  2/π V₀ ≈ 0.637 V₀

A dc meter can be used in an ac circuit if it is connected in the full wave rectifier circuit. The average value of the rectified current is the same as the average current in any half cycle, i.e., 2/π time the maximum current i₀. A more useful way to describe a quantity is the root mean square (rms) value. We square the instantaneous current, take the average (mean) value of i² and finally take the square root of that average. This procedure defines the root-mean-square current denoted as irms. Even when i is negative, i² is always positive so irms is never zero (unless i is zero at every instant). Hence,

Similarly, we get RMS value V_rms = V₀/√2 ≈ 0.707 V₀

The square root of the mean square value is called the virtual value and is the value give by ac instruments.

Thus, when we speak of our house hold power supply as 220 volts ac, this means that the rms voltage is 220 volts and its voltage amplitude is,

            V₀ = √2V_rms = 311 volt

Form Factor

The ratio,       

rms value/average value = (V₀/√2)/(2V₀/π)

                                   = π/2√2 = 1.111

is known as or factor,

Note:

1. The average value of sin wt, cos wt, sin2wt, cos2wt, etc. is zero because it is positive half of the time and negative rest half of the time. Thus,

2. The average value of sin² wt and cos² wt is 1/2

3. Like SHM, general expressions of current/voltage in an sinusoidal ac are,

     = i₀ sin (ωt ± f),                   V = V₀ sin (ωt ± f)

or i = i₀ cos (ωt ± f),        and     V = V₀ cos (ωt ± f)

Ray Optics - Optical Instruments

Optical Instruments

Optical instruments are devices that are used to processes light waves to enhance an image for viewing, or analyze light waves to determine one of a number of characteristic properties. The first optical instrument invented was telescope and was used for magnification of distant images, and microscopes used for magnifying very tiny images. 

Simple Microscope

It is an optical instrument used to see very small objects. It's magnifying power is given by

m = Visual angle with instrument (β)/Visual angle when object is placed at least distance of distinct vision(α)

Simple microscope

(i) It is a single convex lens of lesser focal length.

(ii) Also called magnifying glass or reading lens.

(iii) Magnification's, when final image is formed at D and ∞

     (i.e. m_D and m_∞)    m_D = (1 + D/f)_max

     and       m_∞ = (D/f)_min

(iv) If lens is kept at a distance a from the eye then m_D = 1 + D – a /f and m_∞ = D–a/f

Compound Microscope

(i) Consist of two converging lenses called objective and eye lens.

(ii) f_{eye lens} > f_objective and (diameter)_{eye lens} > (diameter)_objective

(iii) Intermediate image is real and enlarged.

(iv) Final image is magnified, virtual and inverted.

(v) u_o = Distance of object from objective(o), v_o = Distance of image (A'B') formed by objective from objective, u_e = Distance of A'B' from eye lens, v_e = Distance of final image from eye lens, f_o = Focal length of objective, f_e = Focal length of eye lens.

(vi) Final image is formed at D: Magnification m_D = – v_o/u_o (1+ D/f_e) and length of the microscope tube (distance between two lenses) is L_D = v_o + u_e.

Generally object is placed very near to the principal focus of the objective hence u_o = f_o. The eye piece is also of small focal length and the image formed by the objective is also near to the eye piece.

So v_o = L_D, the length of the tube.

Hence, we can write m_D = –L/d_o (1+ D/f_e)

(vii) Final image is formed at ∞: magnification m∞ = –v_o/u_o.D/f_e and length of tube L∞ = v_o + f_e

      In terms of length m∞ = (L_∞ – f_o – f_e)D/f_o f_e

(viii) For large magnification of the compound microscope, both f_o and f_e should be small.

(ix) If the length of the tube of microscope increases, then its magnifying power increases.

(x) The magnifying power of the compound microscope may be expressed as M = m_o × m_e; where m_o is the magnification of the objective and m_e is magnifying power of eye piece.

Astronomical Telescope

By astronomical telescope heavenly bodies are seen.

(i) f_objective >; f_{eye lens} and d_objective >; d_{eye lens}

(ii) Intermediate image is real, inverted and small.

(iii) Final image is virtual, inverted and small.

(iv) Magnification: m_D = –f_o/f_e (1+f_e/D) and m_∞ = –f_o/f_e

(v) Length : L_D = f_o+ u_e and L_∞ = f_o + f_e

Resolving Limit and Resolving Power

Microscope: In reference to a microscope, the minimum distance between two lines at which they are just distinct is called Resolving limit (RL) and it's reciprocal is called Resolving power (RP)

λ = Wavelength of light used to illuminate the object,

µ = Refractive index of the medium between object and objective,

θ = Half angle of the cone of light from the point object,

µ sin θ = Numerical aperture.

Telescope: Smallest angular separations (dθ) between two distant objects, whose images are separated in the telescope is calledresolving limit. So revolving limit dθ = 1.22λ/a and resolving power  where a = aperture of objective.

Refraction of Light through Plane Surfaces

Refraction of Light

The bending of the ray of light passing from one medium to the other medium is called refraction.

(i) The refraction of light takes place on going from one medium to another because the speed of light is different in the two media.

(ii) Greater the difference in the speeds of light in the two media, greater will be the amount of refraction.

(iii) A medium in which the speed of light is more is known as optically rarer medium and a medium is which the speed of light is less, is known as optically denser medium.

(iv) When a ray of light goes from a rarer medium to a denser medium, it bends towards the normal.

(v) When a ray of light goes from a denser medium to a rarer medium, it bends away from the normal.

Refractive Index

(i) Refractive index of a medium is that characteristic which decides speed of light in it.

(ii) It is a scalar, unit less and dimensionless quantity.

(iii) Absolute refractive index:

When light travels from vacuum to any transparent medium then refractive index of medium w.r.t. vacuum is called it's absolute refractive index i.e. _vacuumµ_medium = c/v

Absolute refractive indices for glass, water and diamond are respectively µ_g = 3/2 = 1.5, µ_w = 4/3 = 1.33 and µ_D = 12/5 = 2.4

(iv) Relative refractive index:

When light travels from medium (1) to medium (2) then refractive index of medium (2) w.r.t. medium (1) is called it's relative refractive index i.e. ₁µ₂ = µ₂/µ₁ = v₁/v₂ (where v₁ and v₂ are the speed of light in medium 1 and 2 respectively).

(v) When we say refractive index we mean absolute refractive index.

(vi) The minimum value of absolute refractive index is 1. For air it is very near to 1. ( 1.003)

(vii) Cauchy's equation : µ = A + B/λ² + C/λ⁴ + ...         (1_Red > 1_Violet so m_Red < m_Violet)

(viii) If a light ray travels from medium (1) to medium (2), then ₁µ ₂ =  µ₂/µ₁ =  λ₁/ λ₂ = v₁/v₂

(ix) Dependence of Refractive index:

(a) Nature of the media of incidence and refraction.

(b) Colour of light or wavelength of light.

(c) Temperature of the media : Refractive index decreases with the increase in temperature.

(x) Reversibility of light and refraction through several media:

(A) ₁µ₂ = 1/₂µ₁      (B)   ₁µ₂x₂µ₃x₃µ₁ = 1 or ₂µ₃ = ₁µ₃/₁µ2

Snell's law:

The ratio of sine of the angle of incidence to the angle of refraction (r) is a constant called refractive index

i.e.   sin i/sin r = µ (a constant).

For two media, Snell's law an be written as ₁µ₂ = µ₂/µ₁ = sin i/sin r or µ₁sin i = µ₂sin r

Refraction Through Glass

Lateral Shift:

The refracting surfaces of a glass slab are parallel to each other. When a light ray passes through a glass slab it is refracted twice at the two parallel faces and finally emerges out parallel to it's incident direction i.e. the ray undergoes no deviation δ = 0 . The angle of emergence (e) is equal to the angle of incidence (i)

The Lateral shift of the ray is the perpendicular distance between the incident and the emergent ray, and it is given by MN = t secrsin (i – r)

Apparent Depth

When object is in denser medium and observer is in rarer medium

(i) µ = Real depth/Apparent depth = h/h'

(ii) Real > Apparent depth

(iii) Shift d = h – h' = (1 – 1/µ)h.

(iv) If a beaker contains various immiscible liquids as shown then Apparent depth of

bottom d₁/µ₁ + d₂/µ₂ + d₃/µ₃ + ...

Object is in rarer medium and observer is in denser medium

(i) µ = h'/h

(ii) Real depth < Apparent depth.

(iii) d = (µ = 1)h

Total Internal Reflection (TIR)

When a ray of light goes from denser to rarer medium it bends away from the normal and as the angle of incidence in denser medium increases, the angle of refraction in rarer medium also increases and at a certain angle, angle of refraction becomes 90°, this angle of incidence is called critical angle (C).

When Angle of incidence exceeds the critical angle than light ray comes back in to the same medium after reflection from interface. This phenomenon is called Total internal reflection (TIR).

(i) µ = 1/sin C = cosec C where µ means; _Raserµ_Denser

(ii) Conditions for Total Internal Reflection

(a) The ray must travel from denser medium to rarer medium.

(b) The angle of incidence i must be greater than critical angle C

Ray Optics - Refraction Through Lenses

Lens

(i) Lens is a transparent medium bounded by two refracting surfaces, such that at least one surface is curved. Curved surface can be spherical, cylindrical etc.

(ii) Lenses are of two basic types convex which are thicker in the middle than at the edges and concave for which the reverse holds.

(iii) Principal focus : We define two principal focus for the lens. We are mainly concerned with the second principal focus (F). Thus wherever we write the focus, it means the second principal focus.

First principal focus : An object point for which image is formed at infinity.

Second principal focus : An image point for an object at infinity.

Focal Length, Power and Aperture of Lens

Focal length (f) :  Distance of second principal focus from optical centre is called focal length

                            f_convex –> positive, f_concave –> negative, f_plane –> ∞

Aperture: Effective diameter of light transmitting area is called aperture. Intensity of image ∝ (Aperture)2

Power of lens (P): Means the ability of a lens to deviate the path of the rays passing through it. If the lens converges the rays parallel to the principal axis its power is positive and if it diverges the rays it is negative.

Power of lens P = 1/f(m) = 100/f(cm); Unit of power is Diopter (D)

          P_convex –> positive, P_concave –> negative, P_plane –> zero.

Lens Formula

Lens formula: The expression which shows the relation between u, v and f is called lens formula.

           1/f = 1/v – 1/u

Lens maker formula: If R₁ and R₂ are the radii of curvature of first and second refracting surfaces of a thin lens of focal length f and refractive index µ (w.r.t. surrounding medium) then the relation between f, µ, R₁ and R₂ is known as lens maker formula.        

          1/f = (µ – 1) {1/R₁ – 1/R₂}

Magnification

The ratio of the size of the image to the size of object is called linear magnification.

(i) Transverse magnification : m = I/O = v/u = f/f+u = f–v/f (use sign convention while solving the problem)

(ii) Areal magnification : m_s = A_i/A_o = m² = (f/f+u)²

        (A_i = Area of image, A_o = Area of object)

Relation between object and image speed: If an object moves with constant speed (V_o) towards a convex lens from infinity to focus, the image will move slower in the beginning and then faster.

Also     V₁ = (f/f+u)².V_o

Newton Formula

If the distance of object (x₁) and image (x₂) are not measured from optical centre, but from first and second principal foci then Newton's formula states f² = x₁x₂

Lens Defects

Chromatic aberration: Image of a white object is coloured and blurred because µ (hence f) of lens is different for different colours. This defect is called chromatic aberration.

            µ_v > µ_R so f_R > f_V

Mathematically chromatic aberration = f_R – f_v = ωf_y

ω = Dispersive power of lens.

f_y = Focal length for mean colour √f_Rf_v

Removal: To remove this defect i.e. for Achromatism we use two or more lenses in contact in place of single lens.

Mathematically condition of Achromatism is : ω₁/f₁ + ω₂/f₂ = 0

or     ω₁f₂ = –ω₂f₁

Spherical aberration:

Inability of a lens to form the point image of a point object on the axis is called Spherical aberration.

In this defect all the rays passing through a lens are not focused at a single point and the image of a point object on the axis is blurred.

Removal: A simple method to reduce spherical aberration is to use a stop before and infront of the lens. (but this method reduces the intensity of the image as most of the light is cut off). Also by using plano-convex lens, using two lenses separated by distance d = F – F', using crossed lens.

Spectrum of Light

Spectrum of Light

The ordered arrangements of radiations according to wavelengths or frequencies is called Spectrum. Spectrum can be divided in two parts Emission spectrum and Absorption spectrum.

Emission spectrum : When light emitted by a self luminous object is dispersed by a prism to get the spectrum, the spectrum is called emission spectra.

Continuous emission spectrum

(i) It consists of continuously varying wavelengths in a definite wavelength range.

(ii) It is produced by solids, liquids and highly compressed gases heated to high temperature.

(iii) e.g. Light from the sun, filament of incandescent bulb, candle flame etc.

Line emission spectrum

(i) It consist of distinct bright lines.

(ii) It is produced by an excited source in atomic state.

(iii) e.g. Spectrum of excited helium, mercury vapours, sodium vapours or atomic hydrogen.

Band emission spectrum

(i) It consist of district bright bands.

(ii) It is produced by an excited source in molecular state.

(iii) e.g. Spectra of molecular H₂, CO, NH₃ etc.

Absorption spectrum: When white light passes through a semi-transparent solid, or liquid or gas, it's spectrum contains certain dark lines or bands, such spectrum is called absorption spectrum (of the substance through which light is passed).

(i) Substances in atomic state produces line absorption spectra. Polyatomic substances such as H₂, CO₂ and KMnO₄ produces band absorption spectrum.

(ii) Absorption spectra of sodium vapour have two (yellow lines) wavelengths D₁(5890Å) and D₂(5896 Å)

Refraction Through Prism

Prism Optics

Prism is a transparent medium bounded by refracting surfaces, such that the incident surface (on which light ray is incidenting) and emergent surface (from which light rays emerges) are plane and non parallel.

Prism Refraction

i → Angle of incidence,

e → Angle of emergence,

A → Angle of prism or refracting angle of prism,

r₁ and r₂ → Angle of refraction,

δ → Angle of deviation

A = r₁ + r₂ and i + e = A + δ

For surface AC µ = sin i/sin r₁; For surface AB 1/µ = sin r₂/sin e

Prism Deviation:

For thin prism δ = (µ –1)A. Also deviation is different for different colour light e.g. µ_R < µ_V so δ_R < δ_V.

µ_Flint > µ_Crown so δ_r > δ_c

(i) Maximum deviation: Condition of maximum deviation is

i = 90^o => r₁ = C, r₂ = A – C

and from Snell's law on emergent surface

e = sin^–1 [sin(A–C)/sin C]

δ_max = π/2 + sin^–1 [sin(A–C)/sin C] – A

(ii) Minimum deviation: It is observed if i = e and r₁ = r₂ = r, deviation produced is minimum.

(a) Refracted ray inside the prism is parallel to the base of the prism for equilateral and isosceles prisms.

(b) r = A/2 and i = A + δ_m/2

(c)  (Prism formula)

Prism Dispersion

The splitting of white light into its constituent colors is called dispersion of light.

Angular dispersion (θ): Angular separation between extreme colours i.e. θ = δ_V – δ_R = (µ_V – µ_R)A. It depends upon µ and A.

Dispersive power (ω): ω = θ/δ_y = µ_V –µ_R/µ_y – 1

            where {µ_y – µ_V+µ_R/2}

=> It depends only upon the material of the prism i.e. m and it doesn't depends upon angle of prism A

Wheatstone bridge and Potentiometer

WHEATSTONE BRIDGE

Wheatstone bridge is an arrangement of four resistances which can be used to measure one of them in terms of rest. Here arms AB and BC are called ratio arm and arms AC and BD are called conjugate arms.

Balanced Wheatstone bridge: The bridge is said to be balanced when deflection in galvanometer is zero i.e. no current flows through the galvanometer or in other words V_B = V_D. In the balanced condition P/Q = R/S, on mutually changing the position of cell and galvanometer this condition will not change.

Unbalanced Wheatstone bridge: If the bridge is not balanced current will flow from D to B if V_D > V_B i.e. (V_A – V_D) < (V_A – V_B) which gives PS > RQ.

Applications of Wheatstone bridge: Meter bridge, post office box and Carey Foster bridge are instruments based on the principle of Wheatstone bridge and are used to measure unknown resistance.

METER BRIDGE: In case of Meter Bridge, the resistance wire AC is 100 cm long. Varying the position of tapping point B, bridge is balanced. If in balanced position of bridge AB = l,

BC = (100 – l)

So that .Q/P = (100–l)/l

Also P/Q = R/S ⇒ S = (100–l)/l R

Solved example 1: In Wheatstone bridge P = 9 ohm, Q = 11 ohm, R = 4 ohm and S = 6 ohm. How much resistance must be put in parallel to the resistance S to balance the bridge

(A) 24 ohm      (B) 44/9 ohm          (C) 26.4 ohm         (D) 18.7 ohm

Solution: (C) (For balancing bridge)

⇒ S' = 4×11/9 = 44/9 ⇒ 1/S' = 1/r + 1/6

⇒ 9/44 – 1/6 = 1/r ⇒ r = 132/5 = 26.4 Ω

Solved example 2: A voltmeter having a resistance of 998 ohms is connected to a cell of emf 2 volt and internal resistance 2 ohm. The error in the measurement of emf will be

(A) 4 ×10^–1 volt                                  (B) 2 ×10^–3 volt

(C) 4 ×10^–3 volt                                  (D) 2 ×10^–1 volt

Solution: (C) Error in measurement = Actual value – Measured value

Actual value = 2A

i = 2/998+2 = 1/500 A

Since E = V + ir = ⇒ V = E – ir = 2 – 1/500 × 2 = 998/500 V

Measured value = 998/500 V ⇒ Error = 2 – 998/500 = 4 × 10^–3 volt.

Seebeck Effect & Peltier Effect

Seebeck Effect

Definition: When the two junctions of a thermocouple are maintained at different temperatures, then a current starts flowing through the loop known as thermo electric current. The potential difference between the junctions is called thermo electric emf which is of the order of a few micro-volts per degree temperature difference (µV/°C).

Seebeck Series: The magnitude and direction of thermo emf in a thermocouple depends not only on the temperature difference between the hot and cold junctions but also on the nature of metals constituting the thermocouple.

(i) Seebeck arranged different metals in the decreasing order of their electron density. Few metals forming the series are as below.

Sb, Fe, Cd, Zn, Ag, Au, Cr, Sn, Pb, Hg, Mn, Cu, Pt, Co, Ni, Bi

(ii) Thermo electric emf is directly proportional to the distance between the two metals in series. Farther the metals in the series forming the thermocouple greater is the thermo emf. Thus maximum thermo emf is obtained for Sb-Bi thermo couple

(iii) The current flow at the hot junction of the thermocouple is from the metal occurring later in the series towards that occurring earlier, Thus, in the copper-iron thermocouple the current flows from copper (Cu) to iron (Fe) at the hot junction. This may be remembered easily by the hot coffee.

Variation of thermo emf with temperature: In a thermocouple as the temperature of the hot junction increases keeping the cold junction at constant temperature (say 0°C). The thermo emf increases till it becomes maximum at a certain temperature.

(i) Thermo electric emf is given by the equation E = αβ + 1/2 βt² where  and b are thermo electric constant having units are volt/°C and volt/°C² respectively (t = temperature of hot junction). For E to be maximum (at t = t_n)

             dE/dt = 0 i.e., α + β t_n = 0 ⇒ – α/β

(ii) The temperature of hot junction at which thermo emf becomes maximum is called neutral temperature (tn). Neutral temperature is constant for a thermocouple (e.g. for Cu-Fe, t_n = 270°C)

(iii) Neutral temperature is independent of the temperature of cold junction.

(iv) If temperature of hot junction increases beyond neutral temperature, thermo emf start decreasing and at a particular temperature it becomes zero, on heating slightly further, the direction of emf is reversed. This temperature hot junction is called temperature of inversion (t_i).

(v) Relation between t_n, t_i and t_c is t_n = t_i+t_c/2

Thermoelectric power: The rate of change of thermo emf with the change in the temperature of the hot junction is calledthermoelectric power.

It is also given by the slope of parabolic curve representing the variation of thermo emf with temperature of the hot junction, as discussed in previous section.

The thermo electric power (dE/dt) is also called Seebeck coefficient. Differentiating both sides of the equation of thermo emf with respect to t, we have thermoelectric power

             P = dE/dt = d/dt (αt + 1/2 βt²)

The equation of the thermoelectric power is of the type y = mx + c, so the graph of thermo electric power is as shown.

Peltier Effect

When current is passed through a junction of two different metals, the heat is either evolved or absorbed at the junction. This effect is known as Peltier effect. It is the reverse of Seebeck effect. (When a positive charge flows from high potential to low potential, it releases energy and when positive charge flows from low potential to high potential it absorbs energy).

Peltier coefficient (p)

Heat absorbed or liberated at the junction is directly proportional to the charge passing through the junction i.e. H ∝ Q ⇒ H = πQ; where p is called Peltier coefficient. It’s unit is J/C or volt.

Peltier coefficient of a junction is the amount of heat absorbed or liberated per sec; When 1 amp of current is passed to the thermo couple.

It is found that π = T dE/dT = T × S; where T is in Kelvin and dE/dT = P = Seebeck coefficient S

Appplication of Peltier Effect

Thermoelectric refrigerator : The working of thermoelectric refrigerator is based on Peltier effect.

Kirchoff's Law

Kirchhoff’s Laws

Kirchhoff's first law

This law is also known as junction rule of current law (KCL). According to it the algebraic sum of current meeting at a junction is zero i.e. ∑i = 0.

In a circuit, at any junction the sum of the currents entering the junction must equal the sum of the currents leaving the junction. i₁ + i₃= i₂ + i₄

This law is simply a statement of "conservation of charge".

Kirchhoff second law: This law is also known as loop rule or voltage law (KVL) and according to it "the algebraic sum of the changes in potential in complete traversal of a mesh (closed loop) is zero", i.e. ∑V = 0

(i) This law represents "law of conservation of energy".

(ii) If there n meshes in a circuit, the number of independent equations in accordance with loop rule will be (n –1).

Sign convention for Kirchhoff's law: For the application of Kirchhoff's laws following sign convention are to be considered

(i) The change in potential in traversing a resistance in the direction of current is –iR while in the opposite direction + iR

(ii) The change in potential in traversing an emf source from negative to positive terminal is +E while in the opposite direction –E irrespective of the direction of current in the circuit.

(iii) The change in potential in traversing a capacitor from the negative plate to the positive plate is + q/C while in opposite direction – .

(iv) The change in voltage in traversing an inductor in the direction of current is –L dl/dt while in opposite direction it is +L dl/dt.
 Must Watch 

• Video on Kirchhoff's Law
• Symmetry Analysis on Circuit Network

Tangent's Law & Tangent Galvanometer

Tangent's Law

                     

When a small magnet is suspended in two uniform magnetic fields which are at right angles to each other, the magnet comes to rest at an angle θ with respect to B H . This is called the tangent's law.Tangent Galvanometer is a device based on this law.

 

Prove of Tangent's Law:

• Anticlockwise Torque = mB_H sin(θ)
• Clockwise Torque = mBsin(90-θ)
• In equilibrium position:
• Anticlockwise Torque = Clockwise Torque
• mB_H sin (θ) = mB sin(90-θ)
• B_H tan (θ) = B
• or tan(θ)= B/B_H

Tangent Galvanometer:

It is an instrument which can detect/measure very small electric currents. It is also called as moving magnet galvanometer. It consists of three circular coils of insulated copper wire wound on a vertical circular frame made of nonmagnetic material as ebonite or wood. A small magnetic compass needle is pivoted at the centre of the vertical circular frame. This needle rotates freely in a horizontal plane inside a box made of nonmagnetic material. When the coil of the tangent galvanometer is kept in magnetic meridian and current passes through any of the coil then the needle at the centre gets deflected and comes to an equilibrium position under the action of two perpendicular field : one due to horizontal component of earth and the other due to field set up by the coil due to current (B).

Properties of Magnetic Substances

Magnetic Materials

Types of magnetic materials

On the basis of mutual interactions or behavior of various materials in an external magnetic field, the materials are divided in three main categories.

(i) Diamagnetic materials: Diamagnetism is the intrinsic property of every material and it is generated due to mutual interaction between the applied magnetic field and orbital motion of electrons.

(ii) Paramagnetic materials: In these substances the inner orbits of atoms are incomplete. The electron spins are uncoupled, consequently on applying a magnetic field the magnetic moment generated due to spin motion align in the direction of magnetic field and induces magnetic moment in its direction due to which the material gets feebly magnetized. In these materials the electron number is odd.

(iii) Ferromagnetic materials: In some materials, the permanent atomic magnetic moments have strong tendency to align themselves even without any external field.

These materials are called ferromagnetic materials.

In every unmagnetised ferromagnetic material, the atoms form domains inside the material. The atoms in any domain have magnetic moments in the same direction giving a net large magnetic moment to the domain. Different domains, however, have different directions of magnetic moment and hence the materials remain unmagnetised. On applying an external magnetic field, these domains rotate and align in the direction of magnetic field.

COMPARATIVE STUDY OF MAGNETIC MATERIALS

Property	Diamagnetic substances	Paramagnetic substances	Ferromagnetic substances
Cause of magnetism	Orbital motion of electrons	Spin motion of electrons	Formation of domains
Explanation of magnetism	On the basis of orbital motion of electrons	On the basis of spin and orbital motion of electrons	On the basis of domains formed
Behaviour In a non-uniform magnetic field	These are repelled in an external magnetic field i.e. have a tendency to move from high to low field region.	These are feebly attracted in an external magnetic field i.e., have a tendency to move from low to high field region	These are strongly attracted in an external magnetic field i.e. they easily move from low to high field region
State of magnetisation	These are weekly magnetised in a direction opposite to that of applied magnetic field	These get weekly magnetised in the direction of applied magnetic field	These get strongly magnetised in the direction of applied magnetic field
When the material in the form of liquid is filled in the U-tube and placed between pole pieces.	Liquid level in that limb gets depressed	Liquid level in that limb rises up	Liquid level in that limb rises up very much
On placing the gaseous materials between pole pieces	The gas expands at right angles to the magnetic field.	The gas expands in the direction of magnetic field.	The gas rapidly expands in the direction of magnetic field
The value of magnetic induction B	B < B0	B > B0	B >> B0

Magnetic susceptibility χ	Low and negative |χ| ≈ 1	Low but positive χ ≈ 1	Positive and high χ ≈ 102
Dependence of χ on temperature	Does not depend on temperature (except Bi at low temperature)	Inversely proportional to temperature χ ∝ 1/T  or χ= C/T.This is called Curie law, where C = Curie constant	χ ∝ 1/T-Tc  or χ = C/T-Tc.This is called Curie Weiss law. Tc = Curie temperature
Dependence of χ on H	Does not depend independent	Does not depend independent	Does not depend independent
Relative permeability (μr)	μr < 1	μr > 1	μr >> 1 μr = 102
Intensity of magnetisation (I)	I is in a direction opposite to that of H and its value is very low	I is in the direction of H but value is low	I is in the direction of H and value is very high.
I-H curves
Magnetic moment (M)	The value of M is very low (χ 0 and is in a direction opposite to H.)	The value of M is very low and is in the direction of H	The value of M is very high and is in the direction of H
Transition of materials (at Curie temperature)	These do not change.	On cooling, these get converted to ferromagnetic materials at Curie temperature	These get converted into paramagnetic materials above Curie temperature
The property of magnetism	Diamagnetism is found in those materials the atoms of which have even number electrons	Paramagnetism is found in those materials the atoms of which have majority of electron spins in the same direction	Ferro-magnetism is found in those materials which when placed in an external magnetic field are strongly magnetised
Examples	Cu, Ag, Au, Zn, Bi, Sb, NaCl, H2O air and diamond etc.	Al, Mn, Pt, Na, CuCl2, O2and crown glass	Fe, Co, Ni, Cd, Fe3O4 etc.
Nature of effect	Distortion effect	Orientation effect	Hysteresis effect

where  B₀  is the magnetic induction in vacuum