Magnetic flux and induction

A magnet resting beside a coil does not by itself produce a steady current. Move the magnet and the meter deflects. What changed: field strength, the number of field lines or the circuit’s geometry? We need a quantity that combines these changes.

Rotate a loop in the same field. The field’s contribution through its surface is greatest when perpendicular to the plane; when it lies in the plane, its normal component is zero. Magnetic flux accounts for field, area and surface orientation together. Measure the angle from a chosen normal, an arrow perpendicular to the loop. Thinking of field passing through an area is useful, but no substance is flowing through the opening.

For a flat surface in a uniform field, \Phi=BS\cos\alpha, where α is between B and the chosen normal. B is in teslas and S in m². Doubling area doubles flux; turning to α=90° makes it zero, while at α=0° it equals BS. Cosine appears because we need the normal field component. Do not substitute the angle to the plane, or the angle to current used in the force law: the geometry is different.

Magnetic flux is measured in webers: 1\,\mathrm{Вб}=1\,\mathrm{Тл}\cdot\mathrm{м}^2=1\,\mathrm{В}\cdot\mathrm{с}. Choosing the opposite normal reverses the sign of Φ, without reversing the physical field. Keep the same normal when comparing different times. Flux can be negative by convention, like a velocity component; that does not mean negative area or no field.

Changing B, loop area or tilt can change flux. Electromagnetic induction is the appearance of an EMF when flux through a circuit changes. In a stationary circuit a changing magnetic field causes it; in a moving circuit magnetic force on charges can contribute. Current requires a closed conducting path. Breaking the circuit may stop current without preventing an EMF from arising.

For n turns with equal flux through each, \mathcal E_i=n|\Delta\Phi|/\Delta t gives the magnitude of the average EMF over an interval. ΔΦ is final minus initial flux in webers, Δt is in seconds and n is dimensionless; the answer is in volts. Doubling the number of identical turns doubles the EMF. Zero total flux change means zero signed average EMF, although an EMF may arise and reverse during the interval.

Make the same flux change slowly, then twice as fast. The magnitude of average EMF doubles in the second experiment: both change and time matter. If flux changes linearly, average and instantaneous EMFs agree. Otherwise the instantaneous value can be above or below average. Magnitude of the average and average magnitude differ if direction changes during the interval; the endpoint difference misses such oscillations.

Which way does induced current flow in a closed loop? Lenz’s law says its field opposes the change in the original flux. It contributes an opposing flux during an increase, and supports the original direction during a decrease. If the response reinforced its cause without work, energy could grow by itself. Moving the magnet therefore requires work. The opposition is to the change, not always to the original field itself.

A coil’s current creates its own field. A changing current changes the flux linked to the coil, producing a self-induced EMF. This opposes both increases and decreases in current, tending to preserve the previous state. Inductance L, measured in henries, describes the link. A larger L gives a greater EMF for the same rate of current change. Self-induction requires no motion relative to an external magnet.

One turn encloses flux Φ; n identical turns have total flux linkage nΦ. For a linear coil, n\Phi=LI. Doubling current doubles flux linkage at constant L; for one turn, Φ=LI. This sums contributions over turns rather than representing two different fields. Once nΦ is used, do not multiply by n again: that counts the turns twice. A nonlinear core can make L vary.

Substitute the flux–current relation into the induction law. At constant L, \mathcal E_{\text{си}}=L|\Delta I|/\Delta t. With ΔI in amperes, the result is the magnitude of average EMF in volts. The same current change in half the time gives twice the EMF. Use Lenz’s law for direction. If current rises and falls, the endpoint difference determines the signed average, not the sum of every short voltage spike.

As the source increases current, it works against self-induction. A linear coil stores magnetic energy W=LI^2/2 in joules. Doubling I multiplies energy by four; doubling L at the same I doubles energy. Current sign does not affect it. At I=0 this energy store is absent. As current falls, the field can return energy to the circuit. Self-induction describes energy exchange, not the creation of an extra energy source.

Move a straight rod through a field: force on its moving charges separates them along the rod. If the rod lies along v×B, \mathcal E=Blv\sin\alpha, where α is the angle between velocity and B, l is in metres and v in metres per second. Doubling speed doubles EMF; motion along the field gives zero. For another rod orientation, project the force along it. Without this condition, the formula could predict a voltage between the wrong endpoints.

Uniform rotation makes a loop’s flux vary periodically and generates an alternating EMF. For n identical turns, \mathcal E_{\max}=nBS\omega, with ω in radians per second. Rotating twice as fast doubles both frequency and EMF amplitude. The axis lies in the loop’s plane and is perpendicular to uniform B. Maximum EMF occurs when flux changes fastest, not when flux itself is greatest: these occur at different points in the cycle.

A coil has 100 turns. Flux through each falls uniformly from 0.003 to 0.001 Wb in 0.1 s. Find the EMF magnitude.

Flux changes identically through each of the 100 turns. Find the change for one turn, then multiply by n once.

|ΔΦ| = |0,001 − 0,003| = 0,002 Вб. Flux decreases, but we use the positive magnitude of its change to find the EMF magnitude.

|ε| = n|ΔΦ|/Δt = 100 × 0,002/0,1 = 2 В.

The change is uniform, so 2 V is also the instantaneous EMF magnitude over this interval. In a closed circuit, the induced field would support the decreasing flux.

EMF magnitude: |ε| = 2 В. It is constant over the stated interval because flux decreases uniformly.

EMF is linked to changing flux, which depends on field, area and orientation. Constant flux gives no induced EMF.

Lenz’s law specifies opposition to change and makes induction consistent with energy conservation.

A coil links current to flux linkage and stores LI²/2. Do not count its turns twice.

Next we explore energy exchange in periodic motion.