Magnetic flux. The magnetic flux through a surface measures how many magnetic field lines pass through it. For a uniform field $B$ crossing a flat area $A$, the flux is $\Phi = BA\cos\theta$, where $\theta$ is the angle between $B$ and the area vector (the normal to the surface). Flux is a scalar; its SI unit is the weber (Wb), where $1\,\text{Wb} = 1\,\text{T}\cdot\text{m}^2$. The flux is maximum ($\Phi = BA$) when the field is perpendicular to the surface ($\theta = 0$) and zero when the field lies in the plane of the surface ($\theta = 90^\circ$).
The phenomenon. Faraday and Henry discovered that whenever the magnetic flux linked with a circuit changes, an EMF is induced in it, driving an induced current if the circuit is closed. No battery is needed — the energy comes from whatever changes the flux. The flux can be changed by moving a magnet, changing the field strength, rotating or distorting the coil, or changing its area inside the field.
Faraday's laws of induction. The first law states that an EMF is induced whenever flux changes. The second law is quantitative: the magnitude of the induced EMF equals the rate of change of flux. For a coil of $N$ turns,
- $\varepsilon = -N\frac{d\Phi}{dt}$ — the EMF is $N$ times larger for $N$ turns, because the flux links each turn.
- A faster change in flux gives a larger EMF; a slow change gives a small EMF; no change gives zero EMF.
- The induced current is $I = \varepsilon/R$, where $R$ is the circuit resistance.
Lenz's law and energy conservation. The minus sign in Faraday's law is Lenz's law: the induced current always flows in the direction that opposes the change in flux that produced it. If a magnet's north pole approaches a coil, the coil's near face becomes a north pole to repel it; if it recedes, the near face becomes a south pole to attract it. This is simply conservation of energy — you must do work against the opposing force to keep the flux changing, and that work appears as electrical energy.
Motional EMF. When a straight conductor of length $l$ moves with speed $v$ perpendicular to a magnetic field $B$, the free charges experience a magnetic force and separate, setting up a motional EMF $\varepsilon = Blv$. This is consistent with Faraday's law because the area swept out per second is $lv$, so the flux changes at the rate $Blv$. The mechanical energy spent in moving the rod against the opposing magnetic force becomes electrical energy.
Eddy currents. When the flux through a solid (bulk) conductor changes, induced currents swirl in closed loops within the body — these are eddy currents. They oppose the motion (Lenz's law) and dissipate energy as heat. They are useful in electromagnetic braking, induction furnaces and energy meters, but wasteful in transformer cores — which is why cores are laminated (thin insulated sheets) to break up the eddy-current paths.
A coil of area $0.05\,\text{m}^2$ is placed with its plane perpendicular to a uniform field of $0.4\,\text{T}$. Find the magnetic flux through it. What is the flux if the coil is then turned so its plane is parallel to the field?
Solution- Step 1: When the plane is perpendicular to the field, the area vector is parallel to $B$, so $\theta = 0^\circ$.
- Step 2: $\Phi = BA\cos\theta = 0.4 \times 0.05 \times \cos 0^\circ = 0.02\,\text{Wb}$.
- Step 3: When the plane is parallel to the field, the area vector is perpendicular to $B$, so $\theta = 90^\circ$ and $\cos 90^\circ = 0$.
- Step 4: Hence $\Phi = 0$.
Answer: Flux $= 0.02\,\text{Wb}$ initially, and $0$ when the plane is parallel to the field.
The magnetic flux through a coil of 200 turns changes from $0.01\,\text{Wb}$ to $0.04\,\text{Wb}$ in $0.1\,\text{s}$. Find the average induced EMF.
Solution- Step 1: Change in flux $d\Phi = 0.04 - 0.01 = 0.03\,\text{Wb}$.
- Step 2: Faraday's law (magnitude): $\varepsilon = N\dfrac{d\Phi}{dt}$.
- Step 3: $\varepsilon = 200 \times \dfrac{0.03}{0.1} = 200 \times 0.3$.
- Step 4: $\varepsilon = 60\,\text{V}$.
Answer: The average induced EMF is $60\,\text{V}$.
State Lenz's law and explain how it follows from the conservation of energy.
Solution- Step 1: Lenz's law states that the induced current opposes the change in flux that causes it.
- Step 2: Suppose the induced current instead aided the change — it would increase the flux, which would increase the current further, and so on without any external work.
- Step 3: That would create electrical energy from nothing, violating energy conservation.
- Step 4: So the current must oppose the change, forcing the agent to do work, which appears as electrical energy.
Answer: Lenz's law (induced current opposes the change in flux) is a direct consequence of energy conservation — the opposing force ensures electrical energy comes from mechanical work done.
A metal rod of length $0.5\,\text{m}$ moves at $4\,\text{m/s}$ perpendicular to a uniform magnetic field of $0.3\,\text{T}$. Calculate the motional EMF induced across its ends.
Solution- Step 1: The motional EMF for a rod moving perpendicular to the field is $\varepsilon = Blv$.
- Step 2: Substitute $B = 0.3\,\text{T}$, $l = 0.5\,\text{m}$, $v = 4\,\text{m/s}$.
- Step 3: $\varepsilon = 0.3 \times 0.5 \times 4$.
- Step 4: $\varepsilon = 0.6\,\text{V}$.
Answer: The motional EMF is $0.6\,\text{V}$.
In the previous problem, the rod slides on rails of total circuit resistance $0.2\,\Omega$. Find the induced current and the force needed to keep the rod moving at constant speed.
Solution- Step 1: Induced current $I = \dfrac{\varepsilon}{R} = \dfrac{0.6}{0.2} = 3\,\text{A}$.
- Step 2: The magnetic force on the current-carrying rod opposes its motion: $F = BIl$.
- Step 3: $F = 0.3 \times 3 \times 0.5 = 0.45\,\text{N}$.
- Step 4: To move at constant speed, the applied force must balance this, so it equals $0.45\,\text{N}$.
Answer: Induced current $= 3\,\text{A}$; the force required is $0.45\,\text{N}$.
Why are the cores of transformers and inductors made of thin laminated sheets rather than a solid block of iron?
Solution- Step 1: A changing flux in a solid core induces swirling eddy currents within the metal.
- Step 2: These eddy currents dissipate energy as heat, wasting power and heating the core.
- Step 3: Laminations are thin sheets coated with insulation, which break the large eddy-current loops into small ones.
- Step 4: Smaller loops carry far less current, so eddy-current losses fall sharply.
Answer: Lamination breaks up the eddy-current paths, greatly reducing heat loss and improving efficiency.