Definition: Magnetic Flux
The total number of magnetic field lines passing perpendicularly through a given surface area.
Definition: Electromagnetic Induction
Electromagnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic flux or magnetic field.
Definition: Magnetic Flux
The magnetic flux linked with any surface is equal to the total number of magnetic lines of force passing normally through it.
Or
Magnetic Flux (ΦB) is defined as the total number of magnetic field lines passing normally through a given surface area placed in a magnetic field.
Definition: Faraday's Law of Induction
Whenever the number of magnetic lines of force (magnetic flux) passing through a coil changes, an electric current is induced in the coil. This current is called the induced current.
Define the right-hand thumb rule.
If the current-carrying conductor is held in the right hand such that the thumb points in the direction of the current, then the direction of the curl of the fingers will give the direction of the magnetic field.
Definition: Coefficient of Coupling
The coefficient of coupling K between two coils is the fraction of the total magnetic flux produced by one coil that links with the other coil.
Definition: Mutual Inductance
The property of two coils by which a change in current in one coil induces an emf in the other coil — equal to the magnetic flux linked with one circuit per unit current in the other, or the value of induced emf produced in the secondary circuit per unit rate of change in current in the primary circuit — is called mutual inductance.
OR
Mutual Inductance (M) of a pair of coils is defined as the ratio of the total magnetic flux linkage in the secondary coil to the current in the primary coil that produces it.
Define mutual inductance.
The mutual inductance (M) of two circuits (or coils) is the magnetic flux (Φs) linked with the secondary circuit per unit current (IP) of the primary circuit.
Definition: Eddy Current
The circulating currents induced in conductive materials (bulk pieces of conductors) when the magnetic flux linked with them changes, due to exposure to changing magnetic fields, are called eddy currents.
Definition: Mutual Induction
If we place two coils near each other and pass electric current in one of them, or change the current already passing through it, or stop the current, then an emf is induced in the second coil. This phenomenon of electromagnetic induction is called 'mutual induction'.
Definition: Henry
One henry is the inductance of a circuit in which an emf of 1 volt is induced when the current changes at the rate of 1 ampere per second.
Definition: Coefficient of Mutual Induction
The coefficient of mutual induction of two coils is equal to the number of magnetic flux-linkages in one coil (secondary coil) when a unit current flows in the 'other coil' (primary coil).
Definition: Eddy Currents
Eddy currents (or Foucault currents) are the induced currents set up throughout the volume of a conductor when the magnetic flux linked with it changes.
Definition: Coefficient of Self-Induction
The coefficient of self-induction of a coil is equal to the number of flux-linkages with the coil when unit current is flowing through the coil.
Definition: Self-Induction
The phenomenon of electromagnetic induction in which, on changing the current in a coil, an opposing induced emf is set up in that very coil is called self-induction. The induced emf is called 'back emf'.
Definition: Magnetic Flux
If we consider a plane perpendicular to a uniform magnetic field, then the product of the magnitude of the field and the area of the plane is called the 'magnetic flux' (Ф) linked with that plane.
Definition: Motional emf
When a straight conductor of length 1 moves with a velocity v perpendicular to a magnetic field B an emf (or PD) is developed across the ends of the conductor which is known as motional emf (or PD) given by V = Bvl.
Definition: Electromagnetic Induction
If the circuit is closed, a current flows through it. The emf and the current so produced are called 'induced emf' and 'induced current' and last only while the magnetic flux is changing. This phenomenon is known as 'electromagnetic induction'.
Formula: Magnetic Flux
ΦB = \[\vec B\] ⋅ \[\vec A\] = B A cos θ
| Symbol |
Meaning |
SI Unit |
| \[Φ_B\] |
Magnetic Flux |
Weber (Wb) |
| B |
Magnetic Field Strength |
Tesla (T) |
| A |
Area of the surface |
m² |
| θ |
Angle between B and the normal to the surface |
degrees/radians |
Formula: Magnetic Flux (Flat Surface)
\[\Phi_B=\vec{B}\cdot\vec{A}=BA\cos\theta\]
Formula: Magnetic Flux (Non-Uniform Field or Curved Surface)
For a non-uniform field or a curved surface, flux is calculated by summing contributions over infinitesimally small area elements \[d\vec{A}\]:
\[\Phi_B=\int\vec{B}\cdot d\vec{A}\]
Formula: Flux Through a Coil
For a coil of N turns, each contributing equally:
Φ = N B A cos θ (flux linkage)
Formula: Coefficient of Coupling
M = K\[\sqrt {L_1L_2}\]
Where:
- L1, L2 = Self-inductances of coil 1 and coil 2
- K = Coefficient of coupling (dimensionless, no units)
- Range: 0 ≤ K ≤ 1
Therefore:
M ≤ \[\sqrt {L_1L_2}\]
Formula: Mutual Inductance
N2ϕ21 ∝ I1 ⟹ N2ϕ21 = M ⋅ I1
Therefore:
M = \[\frac{N_{2}\phi_{21}}{I_{1}}\]
Formula: Eddy Current
Eddy current: i = \[\frac {\text {Induced emf (e)}}{\text {Resistance (R)}}\]
Formula: Coefficient of Self-Induction
L = \[-\frac{e}{\Delta I/\Delta t}\]
Units: \[\frac{\mathrm{newton}\times\mathrm{metre}\times\mathrm{second}}{\mathrm{coulomb}\times\mathrm{ampere}}\]
Dimensions: [M L2 T-2A-2].
Formula: Energy Stored in an Inductor
\[U=\frac{1}{2}LI_0^2\]
Formula: Self-Inductance of a Long Solenoid
L = \[\frac{\mu N^2A}{l}=\frac{\mu_r\mu_0N^2A}{l}\]
Formula: Coefficient of Mutual Induction
M = \[-\frac{e_{2}}{\Delta I_{1}/\Delta t}\]
SI units = 'henry' (H) = 1 H = 1V s A-1 = 1 Wb A-1.
Formula: Energy Density
Energy Density = \[\frac {U}{V}\] = \[\frac {1}{2}\]\[\frac {B^2}{μ_0}\]
Formula: Charge Flow During Electromagnetic Induction
q = \[I\times\Delta t=\frac{N}{R}\frac{\Delta\Phi_{B}}{\Delta t}\times\Delta t\] = \[\frac{N}{R}\Delta\Phi_B\]
= \[\frac{\text{number of turns}\times\text{change in magnetic flux}}{\text{resistance}}\]
Formula: Induced Current
I = \[\frac{e}{R}=\frac{N}{R}\frac{\Delta\Phi_B}{\Delta t}\]
Formula: Induced emf
e = \[-N\frac{\Delta\Phi_{B}}{\Delta t}\]
Formula: Magnetic Flux
Ф = (В сos θ) A = BA cos θ
SI units: weber
Dimensions: [ML2T-2A-1]
Formula: Self-Induction
L = \[\frac{N\Phi_B}{I}\]
Faraday's Laws of Electromagnetic Induction
Faraday's First Law
Whenever the magnetic flux linked with a circuit changes, an EMF is induced in the circuit. The induced EMF lasts only as long as the change in flux is taking place.
Faraday's Second Law
The magnitude of the induced EMF in a circuit is directly proportional to the rate of change of magnetic flux through the surface enclosed by that circuit.
It is stated that the direction of induced e.m.f. is always in such a direction that it opposes the change in magnetic flux.
e = `(d phi)/(dt)`
Consider a rectangular metal coil PQRS. Let ‘L’ be the length of the coil. It is placed in a partly magnetic field ‘B’. The direction of the magnetic field is perpendicular to the paper and into the paper. The ‘x’ part of the coil is in the magnetic field at instant t. If the coil is moved towards the right with a velocity v = `dx/dt` with the help of an external agent, such as a hand. The magnetic flux through the coil is:
Φ = BA = BLx
∴ Φ = BLx ...(1)
There is relative motion of a current through the coil. Let ‘i’ be current through the coil.
Three forces act on the coil.
F1 on conductor PL ∴ F1 = Bi x, vertically upward.
F2 on conductor MS ∴ F2 = Bi x, vertically downward.
F3 on conductor SP ∴ F3 = Bi L towards left.
F1 and F2 are equal and opposite and also on the same line. They will cancel each other; F3 is a resultant force. The external agent has to do work against this force.
∴ F3 = −Bi l ...(−ve sign indicates that force is opposite to dx.)
If dx is the displacement in time dt, then the work done (dw) = F3 dx.
∴ dw = − BiL dx
This power is an electrical energy ‘ei’ where ‘e’ is an induced e.m.f.
∴ ei = `-(B_i ldx)/(dt)`
∴ e = `-(BLdx)/(dt)`
∴ e = −BLv
∴ e = `-d/dt (BLx)`
∴ e = `(-d phi)/(dt)` ...[from eq (1)]
Lenz’s Law states that the direction of the induced electromotive force (EMF) and the resulting current in a conductor is always such that it opposes the change in magnetic flux that caused it.
Mathematically, Lenz’s Law is expressed as:
ε = `(-d phi_B)/dt`
Where,
ε = Induced EMF
ΦB = Magnetic flux
The negative sign indicates opposition to the change in flux.
State Faraday’s laws of electromagnetic induction.
First law: Whenever there is a change of magnetic flux in a closed circuit, an induced emf is produced in the circuit. This law is a qualitative law as it only indicates the characteristics of induced emf.
Second law: The magnitude of the induced emf produced in the circuit is directly proportional to the rate of change of the magnetic flux linked with the circuit. This law is known as the quantitative law, as it gives the magnitude of the induced emf.
Faraday’s First Law: Whenever the magnetic flux linked with a circuit changes, an emf is induced in the circuit.
Faraday’s Second Law: The magnitude of the induced emf is equal to the rate of change of magnetic flux.
e = `-(d phi)/dt`
For a coil of N turns:
e = `-N (d phi)/dt`
Negative sign indicates Lenz’s law (direction opposes cause).
Law: Faraday's Second Law or Lenz's Law
Statement:
The direction of the induced emf, or the induced current, in any circuit is such as to oppose the cause that produces it. This law is known as Lenz’s Law.
Explanation / Proof:
- When the north pole of a magnet is moved towards the coil, an induced current flows in the coil in such a direction that the near (left) face of the coil behaves like a north pole.
- Due to the repulsion between the like poles, the motion of the magnet towards the coil is opposed.
- When the north pole of the magnet is moved away from the coil, the induced current flows in such a direction that the near face of the coil becomes a south pole.
- The attraction between opposite poles then opposes the motion of the magnet away from the coil.
In both cases, the induced current opposes the magnet's motion, which is the cause of the current. Therefore, work has to be done to move the magnet, and this mechanical work appears as electrical energy in the coil.
Direction of Induced Current (Fleming’s Right-Hand Rule):
- Stretch the right-hand thumb, forefinger, and middle finger so that they are mutually perpendicular.
- The forefinger points in the direction of the magnetic field.
- The thumb points in the direction of motion of the conductor.
- The middle finger then gives the direction of the induced current.
Conclusion:
Lenz’s Law shows that the induced current always acts in such a direction as to oppose the cause that produces it. This ensures that mechanical energy is converted into electrical energy, and no energy is produced without work being done.
Law: Faraday's First Law or Neumann’s law
Statement:
When the magnetic flux through a circuit is changing, an induced electromotive force (emf) is set up in the circuit whose magnitude is equal to the negative rate of change of magnetic flux. This is also known as Neumann’s Law.
Mathematical Expression:
If ΔΦB is the change in magnetic flux in a time interval Δt, then the induced emf e is given by:
e = \[-\frac{\Delta\Phi_B}{\Delta t}\]
In the limiting case as Δt → 0:
e = \[-\frac{d\Phi_{B}}{dt}\]
- If dΦB is in weber (Wb) and dtdtdt in seconds (s), then the emf eee will be in volts (V).
- This equation represents an independent experimental law, which cannot be derived from other experimental laws.
For a tightly-wound coil of N turns, the induced emf becomes:
e = \[-N\frac{d\Phi_B}{dt}\] or e = \[-\frac{d(N\Phi_B)}{dt}\]
Here, NΦB is called the ‘number of magnetic flux linkages’ in the coil, and its unit is weber-turns.
Explanation:
Consider a magnet and a coil:
- When the north pole of a magnet is near a coil, a certain number of magnetic flux lines pass through the coil.
- If either the coil or the magnet is moved, the number of magnetic flux lines (i.e., the magnetic flux) through the coil changes.
Cases:
- Magnet moved away from the coil → Decrease in magnetic flux through the coil.
- Magnet brought closer to the coil → Increase in magnetic flux through the coil.
In both cases, an emf is induced in the coil during the motion of the magnet.
- Faster motion → Greater rate of change of flux → Higher induced emf.
- If both the magnet and coil are stationary, or both are moving in the same direction with the same velocity, there is no change in flux → No induced emf.
Special Case:
- If the coil is an open circuit (i.e., infinite resistance), emf is still induced, but no current flows.
- This shows that it is the change in magnetic flux that induces emf, not current.
Conclusion:
Neumann’s Law establishes that a changing magnetic flux through a circuit induces an emf, and the induced emf is proportional to the rate of change of flux, with a negative sign indicating the direction (as per Lenz’s law).
Reciprocity Theorem
Statement: The mutual inductance of coil 1 with respect to coil 2 equals the mutual inductance of coil 2 with respect to coil 1.
This is called the Reciprocity Theorem of Mutual Inductance.
Implication: It does not matter which coil drives the current — the mutual inductance M between the pair is always the same property of the system, not just one coil.youtube
Two circular loops, one of small radius r and the other of larger radius R, such that R >> r, are placed coaxially with centres coinciding. Obtain the mutual inductance of the arrangement.
Let a current IP flow through the circular loop of radius R. The magnetic induction at the centre of the loop is
BP = `(mu_0I_P)/(2R)`
As, r << R, the magnetic induction BP may be considered to be constant over the entire cross-sectional area of the inner loop of radius r. Hence magnetic flux linked with the smaller loop will be
`Φ_S = B_PA_S = (mu_0I_P)/(2R)pir^2`
Also, ΦS = MIP
∴ M = `Phi_S/I_P = (mu_0pir^2)/(2R)`
Law: Lorentz Force Law
Statement
When a conducting rod moves in a uniform magnetic field, the charges in it experience a magnetic force (Lorentz force) and a potential difference (motional emf) is produced across its ends. The induced potential difference (motional emf) is given by
V = Bvl
when a straight conductor of length l moves with velocity u perpendicular to a magnetic field B.
Explanation / Proof
-
A conducting rod JK of length l is placed in a uniform magnetic field B (perpendicular to the plane) and moved with constant velocity v perpendicular to its axis.
-
A charge qqq moving with velocity v in magnetic field B experiences a magnetic force of magnitude
Fm = qvB
This force is called the Lorentz force and acts perpendicular to both B and v.
-
Due to this force, electrons move toward end K, making J positive and K negative, so a potential difference V is produced and an electric field E is set up in the rod with magnitude
E = \[\frac {V}{l}\] ...(i)
The electric field exerts an electric force on charge qqq of magnitude
Fe = qE
Conclusion
The induced potential difference across the ends of a straight conductor moving with velocity v perpendicular to a magnetic field B is
V = Bvl
This induced potential difference is called motional emf.
