● Magnetic flux \( Φ_B \) through a surface \( S \) is defined as the surface integral of the magnetic field \( \vec{B} \) over that surface:

\[ \Phi_B = \iint_{S} \vec{B} \cdot d\vec{A} = \iint_{S} B \cos \theta dA, \]

where \( dA \) is an infinitesimal area element with unit normal \( \hat{\mathbf{n}} \), and \( \theta \) is the angle between \( \vec{B} \) and \( \hat{\mathbf{n}} \).

● For a uniform magnetic field \( B \) passing through a flat surface of area \( A \) at an angle \( \theta \):

\[ \Phi_B = B A \cos \theta. \]

● Units: \([Φ_B] = T \cdot m^2 = Wb ext{ (weber)} \]

● Michael Faraday discovered that a changing magnetic flux through a closed conducting loop induces an emf \( \mathcal{E} \) in that loop. Quantitatively:

\[ \mathcal{E} = - \frac{d \Phi_B}{dt}. \]

● The negative sign is a consequence of Lenz’s law, ensuring that the induced emf produces a current whose magnetic field opposes the change in flux (conservation of energy).

● If the loop has \( N \) closely wound turns (coils), the induced emf is

\[ \mathcal{E} = - N \frac{d \Phi_B}{dt}. \]

● If a conductor of length \( L \) moves with velocity \( \vec{v} \) perpendicular to a uniform magnetic field \( \vec{B} \), an emf is induced between its ends:

\[ \mathcal{E} = B L v. \]

● Lenz’s law states: “The induced current always flows in such a direction that the magnetic field it creates opposes the change in the original magnetic flux.”

● This is a direct consequence of energy conservation. For example, if the magnetic flux through a loop is increasing into the page, the induced current circulates to create a magnetic field out of the page, opposing the increase.

● In practice, to determine direction:
1. Identify whether the original flux through the loop is increasing or decreasing.
2. Determine the direction of induced magnetic field needed to oppose that change.
3. Use the right-hand rule (curl of fingers for current, thumb for magnetic field) to find the direction of current in the loop.

● Faraday’s law can be expressed in integral form for an open conducting path or a stationary loop experiencing a changing magnetic field:

\[ \oint_{\mathcal{C}} \vec{E} \cdot d\vec{\ell} = -\frac{d}{dt} \iint_{S} \vec{B} \cdot d\vec{A}, \]

where:
- \( C \) is a closed path (loop).
- \( d\vec{\ell} \) is an infinitesimal element along the loop.
- \( S \) is any surface bounded by \( C \).
- \( \vec{E} \) is the induced electric field (non-conservative, since \( \oint \vec{E} \cdot d\ell \neq 0 \) when \( d\Phi_B/dt \neq 0 \)).

This equation emphasizes that a time-varying magnetic field induces a non-conservative electric field.

● A transformer transfers electrical energy between two or more circuits through electromagnetic induction. It consists of two coils—primary and secondary—wound around a common ferromagnetic core to enhance magnetic coupling (flux linkage).

● Let \( N_p \) be the number of turns in the primary coil, \( N_s \) the number in the secondary. If an alternating current \( I_p(t) \) in the primary produces a time-varying flux \( \Phi_B(t) \) through the core, the induced emf in the primary and secondary are:

\[ \mathcal{E}_p = - N_p \frac{d \Phi_B}{dt}, \quad \mathcal{E}_s = - N_s \frac{d \Phi_B}{dt}. \]

● Ideal transformer assumptions:
- All the magnetic flux produced by the primary links the secondary (i.e., no leakage flux).
- Negligible resistance in the windings (no ohmic losses).
- Infinite core permeability (so that negligible magnetizing current is required).

● From these, the ratio of magnitudes of voltages is:

\[ \frac{V_s}{V_p} = \frac{N_s}{N_p}. \]

● Power conservation (ideal case): \( P_p = P_s \Longrightarrow V_p I_p = V_s I_s. \)

● A generator converts mechanical energy into electrical energy by rotating a coil in a magnetic field, thereby inducing an emf (Faraday’s law). Consider a single loop of wire of area \( A \) rotating with angular speed \( \omega \) in a uniform magnetic field of magnitude \( B \).

● If the plane of the loop makes angle \( \theta(t) = \omega t \) with the magnetic field, the magnetic flux through the loop is:

\[ \Phi_B(t) = B A \cos(\omega t). \]

● The induced emf (magnitude) in the coil (assuming \( N \) turns) is:

\[ E(t) = N \left| \frac{d\Phi_B}{dt} \right| = N B A \omega \left| \sin(\omega t) \right|. \]