IB Physics Unit 11. Electromagnetic Induction: Notes

Understandings

➔ Electromotive force (emf)

➔ Magnetic flux and magnetic flux linkage

➔  Faraday’s law of induction

➔ Lenz’s law

Applications and skills

➔ Describing the production of an induced emf by a changing magnetic flux and within a uniform magnetic field

➔ Solving problems involving magnetic flux, magnetic flux linkage, and Faraday’s law

➔ Explaining Lenz’s law through the conservation of energy

Equations

➔ Flux: Φ = BA cosθ

➔  Faraday’s / Neumann’s equation: ε = –N (dΦ/dt)

➔ emf induced in moving rod: ε = Bvl

➔ in side of coil with N turns: ε = BvlN

11.1 Electromagnetic Induction

Electromagnetic induction: When an electric charge moves in a magnetic field, then a force acts on it. In a reverse sense, a movement or change in magnetic field relative to stationary charge gives raise to an electric current.

Induced emf (ε)

  • Definition: Potential difference generated by electromagnetic induction.

For a rod of length L moved with velocity v in a region of magnetic field B:

  • If ​the rod moves from left to right, and thus, its electrons move perpendicular to the magnetic field, they experience a downward force along the rod and an electric field is established.

    FieldMagForce.png

  • Flow of electrons quickly stops due to electrostatic repulsion at the bottom, and thus, the current exists for a short period of time.

  • Without movement, emf is not induced.

  • Formula if the rod is moved connected to wires (the work done to separate electrons leads to an induced emf): ε = BvL.

Magnetic flux (Ф)

  • Definition: “Product of the magnitude of the normal component of magnetic field strength and area through which it passes.”

  • Intuitive picture: Number of magnetic field lines crossing a certain area.

  • Formula: Ф = BAcosθ, where A is the area and θ is the angle between the magnetic field strength direction and the direction normal to the loop area.

  • Units: weber (Wb)

FieldArea.png
  • Definition for a rod: “Product of magnitude and the rate at which the area swept out by the rod is changing” = ∆Ф/∆t.

FieldSmth.png
  • Magnetic flux linkage: Magnetic flux multiplied by the N turns in a loop. Ф = NBAcosθ.

  • Magnetic flux density: numerically equivalent to magnetic field strength.

    • Induced emf = magnetic flux density x rate of change of area =​ B∆A/∆t.

Faraday’s Law

  • Definition: “Induced emf is equal to the negative rate of change of magnetic flux linkage.”

    • Negative sign exists due to Lenz’s law (see below).​

  • Formula: ε = -N∆Ф/∆t.

Magnetic field away from viewerwiremagneticfield.png

Coil of area A with N turns

Example of Faraday’s Law:

A coiled wire is moving into a magnetic field.

The induced current should create a magnetic field in a direction opposed to the existing field (in this case, opposing “away from”, therefore, towards the viewer)

  • Rod (perpendicular to field): in time ∆t, a rod of length L will move a distance s = v∆t, cutting magnetic field lines as it moves in the magnetic field. A = Ls

    • Formula: ​∆Ф = ∆BAcos0º = ∆BA = ∆BLs = BLv∆t, and hence, ε = BvL.

Lenz’s Law

  • Definition: “The induced emf will be in such a direction to oppose the change in the magnetic flux that crea

  • ted the current​”. It is equivalent to energy conservation.

  • Work done by magnetic forces that arises due to current is dissipated as thermal energy.

Examples:

  • Rod: Force in the rod must oppose 

    • Use left-hand rule twice: Firstly to find the direction of the current in the loop. Secondly, to find the force induced on the rod due to the current the motion. Hence, if it moves towards the right, a leftwards force will appear indicating a counter-clockwise induced current. ​

FluxDensity.png

  • Loop wire and a wire with increasing current: Magnetic flux is increasing into the page. Hence, to oppose the increase in magnetic flux (inside the loop), a magnetic field out of the page must exist, and thus, a counter-clockwise current is induced.​

wireloop.png

 
  • Bar magnet through a loop of wire:

    • When approaching the loop, magnetic flux is increasing, and thus, magnetic field must oppose the increase, with a counter-clockwise current.

    • When leaving the loop, the magnetic flux is decreasing, and the current is now clockwise. ​

7.jpeg

The opposite magnet (south pole first) would have the exact opposite effect.

ELECTROMAGNETIC INDUCTION

THE EXPERIMENTS OF FARADAY AND HENRY  

The discovery and understanding of electromagnetic induction are based on a long series of experiments carried out by Faraday and Henry. These experiments are illustrated by the following figures.
When the bar magnet is pushed towards the coil, the pointer in the galvanometer G deflects.
Current is induced in coil C
There is also another law for finding the direction of induced current. This is Fleming’s right hand rule. According to this rule, if we stretch the right-hand thumb and two nearby fingers perpendicular to one another such that the first finger points in the direction of magnetic field and the thumb in the direction of motion of the conductor, then the middle finger will point in the direction of the induced current.
   
Total flow of charge due to change of flux (Δφ):

METHODS OF INDUCING E.M.F.

As is known, e.m.f. is induced in a circuit only when amount of magnetic flux linked with the circuit changes. As φ = BA cos θ, therefore three methods of producing induced e.m.f. :
  1. By changing B
  2. By changing A
  3. By changing θ (orientation of the coil)
When a conductor of length l moves with a velocity v in a magnetic field of strength B so that magnetic flux linked with the circuit changes, the e.m.f. induced (ε) is given by
ε = B l v.

INDUCED E.M.F. AND ITS DIRECTION

Case (i) In conducting rod
The induced e.m.f. is generated  because of rotation of a conducting rod in a perpendicular magnetic field
 also, e = – BAf
where f = frequency of rotation and
A = πr2, where r is the radius of circle in which this rod moves, hence r = l.
ω = angular velocity, l = length of conducting rod.

 

Case (ii) In disc
Induced e.m.f generated in a disc rotating with a constant angular velocity in a perpendicular magnetic field
where  A = area of disc = πr2, r = radius of disc,
ω = angular velocity of disc.

 

Case (iii) In two coils
When two coils are arranged as shown in the figure
  • if key K is closed then current in P will flow in clockwise direction and consequently induced current in Q will flow in anticlockwise direction. (see fig.a)
  • when key K is opened then current in P falls from maximum to zero and consequently induced current in Q will flow in clockwise direction. (see fig.b)

 

Case (iv) In three coils arranged coaxially
Three coils P, Q and R are arranged coaxially as shown in figure. Equal currents are flowing in coils P and R . Coils Q and R are fixed. Coil P is moved towards Q. The induced current in Q will be in anti-clockwise direction so that it may oppose the approach of P according to Lenz’s law. As the face of P towards Q is a south pole hence plane of Q towards P will also be a south pole.
As there is no relative motion between Q and R, hence no current is induced in Q due to R.

 

Case (v) Current increases in straight conductor
When current in the straight conductor is increased then
  • the direction of induced current in the loop will be clockwise so that it may oppose the increase of magnetic flux in the loop in downward direction.
  • the direction of induced current in the loop will be anti-clockwise so that it may oppose the increase of magnetic flux in the loop in upward direction.
Case (vi) Magnet dropped freely in long vertical copper tube
The resistance of copper tube is quite negligible and hence maximum induced current are generated in it due to the motion of the magnet. Due to these induced current the motion of magnet is opposed to maximum. Consequently the acceleration of the magnet will be zero (a = g – g = 0).

 

Case (vii) Magnet dropped freely into a long solenoid of copper wire
The resistance of copper solenoid is much higher than that of copper tube. Hence the induced current in it, due to motion of magnet, will be much less than that in the tube. Consequently the opposition to the motion of magnet will be less and the magnet will fall with an acceleration (a) less than g. (i.e. a < g).

 

Case (viii) Motional EMF
Induced emf in a conducting rod moving perpendicular through a uniform magnetic field as shown
The induced emf produced across the rod
This is also called motional emf and it develops when a metal rod cuts magnetic lines of force.
Special case :   If the rod moves in the magnetic field making an angle θ with it, then induced emf .
COMMON DEFAULT
🗴 Incorrect.  When there is no change in magnetic flux no induced current is produced.
✓ Correct. Consider the case (viii) discussed above. There is no change in the magnetic flux through the rod, still induced emf is produced.

 

Case (ix) A straight conductor (slider) moving with velocity v on a U shaped wire placed in a uniform magnetic field.
The induced current produced is  

 

Case (x) When a rectangular loop  perpendicular to the magnetic field is pulled out, then forces and being equal and opposite cancel out.

 

Power required to move the loop out
P= F2 × v
Case (xi) The magnet is stationary and the loop is moving towards the magnet.
The induced emf or current I is shown which is in accordance to Lenz's law. In this case the magnetic force causes the charge to move. We know that if a charged particle is in motion in a field it experiences a magnetic force. This is because when charged particle moves it creates its own magnetic field which interacts with the existing magnetic field.

 

Case (xii) The magnet is moving towards the loop which is stationary.
The induced emf or current I is shown which is in accordance to Lenz’s law.  Here the varying magnetic field at the location of loop (due to the movement of magnet) creates an electric field.
We should remember certain points regarding the induced electric field produced due to changing magnetic field.
  • Induced electric field lines form closed loops (different from the electric field lines used to depict electric field produced due to  charges)
  • Induced electric field is non-conservative in nature (again a difference from the electric field produced by electric charges)
Mathematically,
Note:
  1. An emf is induced in a circuit where the magnetic flux is changing even if the circuit is open. But obviously no current will flow. If we close the circuit, the current will start flowing.
  2. In a loop moving in a uniform magnetic field, when the loop  remains in the field, the net emf induced is zero.
         

EDDY CURRENTS

The induced circulating currents produced in a metal itself due to change in magnetic flux linked with the metal are called eddy currents. These currents were discovered by Foucault, so they are also known as Foucault Currents.
The direction of eddy currents is given by Lenz’s law.
Eddy currents produced in a metallic block moving in a non-uniform magnetic field is shown in fig.

APPLICATIONS OF EDDY CURRENT

Like friction, eddy currents are helpful in some fields and have to be increased, while in some other fields they are undesirable and have to be minimised.
  • Dead beat galvanometer
  • Energy meter
  • Speedometer
  • Electric brakes
  • Single phase AC motor
  • Induction furnace
  • Diathermy

 

Note:- In a moving coil galvanometer, damping is necessary to avoid oscillation of display needle. This is brought into practice with the help of eddy currents. The winding of the coil of galvanometer is done on a metallic frame. When the coil rotates the magnetic flux linked with the metallic frame changes due to which eddy currents are developed which oppose the rotation of the coil. This is called dead beat galvanometer.

SELF INDUCTANCE AND MUTUAL INDUCTANCE

SELF INDUCTANCE

The property of a coil by virtue of which the coil opposes any change in the strength of the current flowing through it, by inducing an e.m.f. in itself is called self inductance.
When a current I flows through a coil, the magnetic flux φ linked with the coil is φ = LI, where L is coefficient of self inductance of the coil.
On differentiating, we get
If dI / dt = 1; L = – e.
Hence coefficient of self inductance of a coil is equal to e.m.f. induced in the coil when rate of change of current through the same coil is unity. Coefficient of self induction of a coil is also defined as the magnetic flux linked with a coil when 1 ampere current flows through the same coil.
The value of L depends on geometry of the coil and is given by
where l is length of the coil (solenoid), N is total number of turns of solenoid and A is area of cross section of the solenoid.
The S.I. unit of L is henry. Coefficient of self induction of a coil is said to be one henry when a current change at the rate of 1 ampere/sec. in the coil induces an e.m.f. of one volt in the coil.

 

KEEP IN MEMORY
  1. Energy stored in a coil  (inductor) =
where L is the self-inductance and i current flowing through the inductor.
The energy stored in the magnetic field of the coil.
  1. The self inductance is a measure of the coil to oppose the flow of current through it. The role of self-inductance in an electrical circuit is the same as that of the inertia in mechanics. Therefore it is called electrical inertia.
  2. The magnetic energy density (energy stored per unit volume) in a solenoid

MUTUAL INDUCTANCE

Mutual induction is the property of two coils by virtue of which each opposes any change in the strength of current flowing through the other by developing an induced e.m.f.
Coefficient of mutual inductance (M) of two coils is said to be one henry, when a current change at the rate of 1 ampere/sec. in one coil induces an e.m.f. of one volt in the other coil. The value of M depends on geometry of two coils, distance between two coils, relative placement of two coils etc.
The coefficient of mutual inductance of two long co-axial solenoids, each of length l, area of across section A, wound on an air core is … (1)
where N1 and N2 are total number of turns of the two solenoids.
The mutual inductance M is defined by the equation
N2φ2 = MI1
where I1 is the current in coil 1, due to which flux φ2 is linked with each turn of secondary coil.
Now we can calculate, e.m.f. e2 induced in secondary by a changing current in first coil. From Faraday‘s law
If  ...(2)
The two definitions for M defined by equations (1) and (2) are equivalent. We can express these two equations in words as :
  • M is numerically equal to the flux-linkage in one circuit, when unit current flows through the other. (we use this definition to calculate M)
  • M is numerically equal to the e.m.f. induced in one circuit, when the current changes in the other at the rate of one ampere in each second. (it is used to describe the mutual behavior of two circuits).
For a pair of coils, M12 = M21 = μ0 N1 N2 A/, when wound on one another.

 

KEEP IN MEMORY
  1. Coefficient of self inductance of two coils in series
The effective self inductance is Ls = L1 + L2
If M is the coefficient of mutual inductance between the two coils when they have flux linkage in the same sense,then L = L1 + L2 + 2M
And for flux linkage in opposite direction
L = L1 + L2 – 2M
  1. Coefficient of self inductance of two coils in parallel
  • The coefficient of coupling between two coils having self inductance L1 & L2 and coefficient of mutual inductance M is
  • Generally the value of K is less than 1.
  • If K is 1, then the coupling of two coils is tight while if K < 1, then coupling is loose.
    • Inductance is pure geometrical factor, and is independent of current or applied e.m.f.
    • If the angle between the axis of two closely placed coil is θ then

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