Electromagnetic Induction

 Introduction

  1. Electromagnetic induction

    Electromagnetic induction in the phenomenon of generating an e.m.f by changing the number of magnetic lines of force associated with a circuit. The e.m.f so generated is called induced e.m.f and the corresponding current is called induced current.

  2. Magnetic flux (\phi)

    The number of magnetic lines of force crossing a surface normally is called magnetic flux linked with the surface.

    \phi = \overrightarrow{B}. \overrightarrow{A} = BA \cos \theta

    Where ‘B’ is strength of magnetic field, ‘A’ is the area of the surface and is ‘θ’ the angle which normal to the surface area makes with the direction of the magnetic field.

    The SI unit of \phi is Weber. The cgs unit is Maxwell.

    1 Weber= 10^8 Maxwell. Also, Weber =1 Tesla meter^2

    If a coil has N turns, then

    \phi =NBA \cos θ

  3. Case I: If the magnetic field \overrightarrow{B} is normal to the plane of coil then,

    θ=0^{\circ}, \phi = NBA \cos 0^{\circ} = NBA (Maximum)

    Case II: If \overrightarrow{B} s parallel to the plane of the coil the θ= 90^{\circ}

    θ=90^{\circ}

    ∴ \phi =0 (Minimum)

    The magnetic flux can be +ve, -ve or zero depending on the angle θ. Theflux is +ve for 0^{\circ}< θ>90^{\circ} and -ve for180^{\circ} > θ>90^{\circ} $

    The magnetic flux is taken as -ve if field lines enter the area and +ve if field lines leave the area.

    The magnetic flux through a closed surface is always zero i.e.

    \oint \overrightarrow{B}. \overrightarrow{dA} = 0

    This is called Gauss’s theorem for magnetostatics.


Faraday’s law of Electromagnetic induction

  1. First law

    Whenever there is change in the magnetic flux linked with a coil, an induced e.m.f is produced in the coil. There is an induced current only when coil circuit is complete (i.e closed). The induced e.m.f lasts so long as the change in flux takes places.

  2. Second law The magnitude of e.m.f induced in the circuit is directly proportional to the rate of change of magnetic flux linked in the circuit i.e

    E \propto \dfrac{d \phi}{dt} = \dfrac{\phi_2 - \phi_1}{t} = - \dfrac{d\phi}{dt}

    \phi = NBA \cos \theta = NBA \cos \omega t

    E= - \dfrac{d}{dt} (NBA \cos \omega t) =

  3. Cases

    If area A alone is changing, then

    E=-NB cosωt(dA/dt)

    If the magnetic field B alone is changing, then

    E=-NA cosωt(dB/dt)

    If θ alone is changing

    ∴E=-NBA cosωt ((dcosωt))/dt =-NBAωsinωt

  4. Thus, induced e.m.f depends on

    i. Number of turns (N)

    ii. Area of coil (A)

    iii. Angular velocity (ω) of the coil

  5. Emf induced in the coil if B, or A or θ is changing.

    Emf induced in the coil doesn’t depend on resistance of coil.

    Emf induced in the coil depends on relative motion of coil and electromagnet system i.e. it the coil or magnet is brought with the high speed towards than more emf will be induced in the coil and vice-versa.

    Emf induced \proptospeed \propto 1/time

  6. Current Induced in the coil (i)

    Current induced i =\dfrac{E}{R}

    i = \dfrac{d \phi}{dt} \dfrac{1}{R}

    Current induced in the coil depends on the resistance of the coil and relative motion of coil and electromagnetic system.

    ∴current induced(i)∝\dfrac{1}{R} \propto speed \propto 1/(time taken)

  7. Charge induced (q)

    q= idt = \dfrac{d \phi}{dt} \dfrac{1}{R} dt

    i = \dfrac{d \phi}{R}

    Charge induced in the coil depends on resistance of coil and in independent of relative motion of coil and electromagnet system i.e., even a coil or magnet brought with high speed or less speed, amount of charge flow in the coil will be constant.


Len’s Law

  1. The direction of induced e.m.f. is given by Lenz’s law.

  2. According to this law, the direction of induced e.m.f. in a circuit is always such as to oppose the change in magnetic flux responsible to it.

  3. Len’s law is in accordance with the principle of conversation of energy. In fact, work done in moving the magnet w.r.t, the coil change into electric energy producing induced current.

  4. The direction of induced current

    This is given by Fleming’s Right-Hand Rule. According to this rule, if we stretch the first figure, the central finger and thumb of our right hand is mutually perpendicular direction, such that first finger points along the magnetic field and thumb points along the direction of motion of conductor, then central figure would give the direction of induced current +ve terminal of battery.

  5. Right Hand Palm Law

    If four finger point the direction of \overrightarrow{B} and thumb points the direction of velocity then palm gives the +ve terminal of battery/direction of induced current. [Fleming’s Left Hand Law gives the direction of force ]


Conducting rod moving in uniform magnetic field

  1. When a conducting rod of length ‘l’ moves with the velocity ‘v’ in uniform magnetic field of induction ‘B’ such that ‘l’ makes angle θ with v, then the magnitude of the average induced emf is given by

    E=vBlsinθ=B (\overrightarrow{l} \times \overrightarrow{v})

  2. Emf induced in the rod will be zero if out of B, l and v any one parallel to other

  3. To move rod with constant speed a= 0

    F_{ext}-F_m=m \times 0= 0

    F_{ext}=F_m

    F_{ext}=BIl=F_m =B(\dfrac{E}{R})l=B(\dfrac{Bvl}{R})l

    F_{ext}=(B^2 l^2 v)/R

  4. Mechanical power applied to pull the rod is given by P=(work done )/(time )

    = F*V

    ∴P= (B^2 l^2 v^2)/R


Conducting rod rotating with angular velocity ω in a uniform magnetic field.

  1. When a rod of length ‘l’ rotates with angular velocity ω in a uniform magnetic field B, the induced emf across the ends of rotating rod is:

    E=1/2 Bωl^2=1/2 B2πfl^2=Bπfl^2=BAf

    ∴E=(Bωl^2)/2=BAf

    Where A=πl^2) is area swept by the rod in the rotation and f is the frequency of rotation.

  2. EMF induced in a close loop moving in a magnetic field

    Emf induced in the rod Blv can be used only for straight conductor.

    If a zig-zig wire or rod or loop is moving then, E= Bvl_{\text{displacement}}

    If a closed loop of any shapr and size moving inside magnetic field then emf induced is closed loop will be zero provided that no load is applied across the closed-loop. (Since l_\text{displacement} of closed loop is zero)

    Once load is applied, cell will be in parallel, thus total emf induced will be 2rBv.


Eddy Current

  1. Eddy currents are the currents induced in the body of a conductor when the amount of magnetic flux linked with the conductor changes.

  2. For example, when we move a metal plate out of a magnetic field, the relative motion of the field and the conductor again induces a current in the conductor. The magnitude of eddy current is

    i=(induced emf)/resistance=E/R=-(d∅/dt)/R

  3. The direction of eddy current is given by Lenz’s law or Fleming’s right-hand rule.

  4. Eddy current are produced when a metal is kept in a varying magnetic field.

  5. Eddy current are basically the current induced in the body of a conductor due to change in magnetic flux induced with the conductor.

  6. Application of Eddy currents

    i. Electromagnetic damping is designing dead beat galvanometers.

    ii. In induction furnace to melt the substance.

    iii. As electromagnetic brakes in controlling the speed of electric trains.

    iv. In induction AC motors.

    v. In speedometers of automobiles and energy meters.

    vi. Eddy current are also used in deep heat treatment of the human body.

  7. Undesirable Effects

    i. They oppose the relative motion.

    ii. They involve loss of energy in the form of heat.

    iii. The excessive heating may break the insulation in the appliances and reduce their life.


Self-Induction

  1. In the property of a coil by virtue of v\ which the coil opposes any chance in the strength of current flowing through it by inducing an e.m.f in itself. ∴ ∅=LI

    Where L is a constant called coefficient of self-induction or self-inductance of the coil.

    For N turns of coil,∅=NLI

  2. The emf induced in the coil is

    E=-(d∅)/dt=(-d)/dt (LI)=-L dI/dt

    ∴=-L dI/dt

    where E is the back e.m.f

  3. The S.I unit of L is Henry or Weber/Ampere.


Energy Stored in Inductor

  1. Energy density=(energy )/volume=1/2 B^2/μ_0

    ∴Energy stored by magnetic field

    =B^2/(2μ_0 ).Volume

  2. An inductor stores energy in the form of magnetic energy given by E=1/2 LI^2 For Solenoid, B=μ_0 nI

    E=(μ_0 nI)^2/(2μ_0 )* volume =1/2 (μ_0 n^2.V)I

    E= 1/2 LI^2


Mutual Induction

  1. It is the property of the two coils by virtue of which each coil opposes any changes in the strength of current flowing through the other by developing an induced e.m.f.

  2. When a current I flowing in the primary coil produced magnetic flux ∅ in the secondary coil, the \phi \propto I

    ∴ ∅=MI

    where M is constant called the coefficient of mutual induction or mutual induction

  3. The SI unit of M is Henry.

  4. The emf induced in the neighboring coil in given by

    E=(-d∅)/dt=-M dI/dt

  5. The value of M depends on

    i. Geometry of two coil i.e size, shape, number of turns, and nature of material on which the coils are wound.

    ii. Distance between the two coils.

    iii. Relative placement of two coils.

  6. Mutual inductance of two coupled coils having self-inductancesL_1 and L_2 is

    M=K \sqrt{L_1 L_2}

    where k is constant called coefficient of coupling between the two coils.

    When two coils are lightly coupled, then K=1

    M=K \sqrt{L_1 L_2} (Maximum)

    For loose coupling K<1 and M decreases

    K=0 for uncoupled coils

  7. K=(magnetic flux lined with secondary coil)/(magnetic flux with primary coil)

  8. the coefficient of mutual inductance of two long co-axial solenoids, each of length l, area of cross-section A, wound on air core is,

    M=(μ_0 N_1 N_2 A)/l

    Where, N_1 N_2 are total number of turns of the two solenoids.

  9. If the solenoid core is of material of permeability μ_0 then

    M=(μ_0 μ_0 N_1 N_2 A)/l


Combination of Inductors

  1. Series combination of inductances

    When two inductors of self-inductances of self-inductance L_1, L_2 are kept far apart then their mutual inductance M=0 Then, the equivalent is:

    L_s=L_1 +L_2

    If two coils are connected in series close to each other them

    L_s=L_1 +L_2+2M (If current in two coils is in the same direction)

    L_s=L_1 +L_2-2M (if current in two coils is in the opposite direction)

  2. If two coils of inductances L_1 and L_2 are connected in parallel, the effective inductance is given by

    L_p=(L_1 L_2-M^2)/(L_1 +L_2+2M) (If current is two coils is in the same directions)

    L_p=(L_1 L_2-M^2)/(L_1 +L_2-2M) (If current in the two coils is in different direction)

    L_p=(L_1 L_2)/(L_1 +L_2) (If the coils are apart)

    i.e 1/L_p =1/L_1 +1/L_2


Transformer

  1. Its works on the principle of mutual inductance and is used in AC only.

  2. It is an electrical device, which is used for changing alternating voltages.

  3. A transformer consists of a primary coil of turns N_p and a secondary coil of turns N_s and a laminated soft iron core.

  4. If E_p andE_s denote the voltage across the primary coil and the secondary coil respectively, then

    E_s/E_p =I_p/I_p =N_s/N_p =K

    Where K is a constant called ‘Transformation Ratio’

  5. For a step-up transformer, K>1, and for a step-down transformer K<1.

    In transformer in practice

    Output power < Input power

  6. Step up transformer

    N_s >N_p \to E_s>E_p, k>1 and I_s<I_p

  7. Step down transformer

    N_s <N_p, so E_s<E_p, k<1 and I_s<I_p

  8. power is lost during working of a transformer is because of following ways,

  9. Flux/Coupling loss due to imperfect coupling.

  10. Copper loss as heat in copper wire.

  11. Eddy loss due to generation of current in iron core.

  12. Hysteresis loss during cycle of magnetization.

  13. Humming loss as sound due to vibration.

  14. Eddy loss is minimized by using laminated plates pasted together with non-conducting glue.

  15. Efficiency of transformer

    η=\dfrac{\text{output power}}{\text{input power}} \times 100 \% =\dfrac{E_s I_s}{E_p I_P} \times100\%

    Thus, a transformer actually transforms power.

    For ideal transformer,

    Output power=Input power

    i.e., E_s I_s=E_p I_P, Also, L \propto N^2

    ∴ E_p/E_s =N_P/N_s =I_s/I_P = \sqrt{ \dfrac{L_P}{L_s}}

  16. Transformer works only in AC circuits not in DC circuits. If DC is connected to primary coil (input) then output voltage is zero.

  17. Frequency is not affected by transformer. It remains constant.

  18. The main use of transformer is in transmission of AC over long distance at extremely high voltages.

Electromagnetic Induction
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