# Electromagnetic Induction

Physics II · Electromagnetism · https://tryals.app/learn/physics-ii/electromagnetic-induction

## A Changing Flux Makes a Field

**Faraday's law** is the first equation in which the electric field stops being electrostatic:

$$\mathcal{E} = -\frac{d\Phi_B}{dt}, \qquad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

where $\Phi_B = \int \mathbf{B}\cdot d\mathbf{A}$. A changing magnetic flux drives an EMF round a loop, and the induced $\mathbf{E}$ has non-zero curl, so no scalar potential describes it, and $\oint\mathbf{E}\cdot d\mathbf{l}$ is no longer zero.

The minus sign is **Lenz's law**: the induced current opposes the change producing it. This is not an extra postulate but a requirement of energy conservation. If the induced current reinforced the change, a small disturbance would amplify itself without limit and generate energy from nothing.

There are three ways to change the flux, and all appear in real machines: change $B$, change the area, or rotate the loop. The last is how a generator works, giving $\mathcal{E} = NBA\omega\sin\omega t$.

**Inductance** quantifies flux linkage per unit current. **Mutual inductance** couples two circuits, $\mathcal{E}_2 = -M\,dI_1/dt$, which is what a transformer exploits. **Self-inductance** gives $\mathcal{E} = -L\,dI/dt$, opposing changes in a circuit's own current, the reason a switched-off inductor can produce a large spark.

Establishing a current stores energy in the field:

$$U = \tfrac{1}{2}LI^2, \qquad u = \frac{B^2}{2\mu_0}$$

exactly parallel to the electrostatic $\tfrac{1}{2}CV^2$ and $\tfrac{1}{2}\varepsilon_0E^2$.

> **Common pitfall:** thinking the induced current opposes the *field*. It opposes the **change** in flux. A loop in a decreasing field carries current that tries to *maintain* the flux, reinforcing the field, not fighting it.

## Practice questions

7 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A loop sits in a magnetic field that is steadily decreasing. Which way does the induced current flow?

A. In the direction that further accelerates the decrease in flux
B. In the direction that opposes the existing magnetic field vector
C. In the direction that maintains the existing flux through the loop
D. No current flows, because the external magnetic field is decreasing

**Answer:** C. In the direction that maintains the existing flux through the loop

**Why:** Lenz’s law opposes the *change*. With the flux falling, the induced current flows so as to maintain it, reinforcing the field rather than fighting it. Only an increasing flux produces an opposing current.

Page: https://tryals.app/practice/physics-ii/electromagnetic-induction/a-loop-sits-in-a-magnetic-field-that-is-steadily-decreasing-which

### 2. The minus sign in Faraday’s law is required by conservation of energy.

**Answer:** True

**Why:** True, with the opposite sign, an induced current would strengthen the change that produced it, amplifying without limit and generating energy from nothing. Lenz’s law is energy conservation expressed as a direction.

Page: https://tryals.app/practice/physics-ii/electromagnetic-induction/the-minus-sign-in-faradays-law-is-required-by-conservation-of-energy

### 3. An inductor of 0.40 H carries a current of 3.0 A. Compute the stored magnetic energy in joules, to one decimal place.

**Answer:** 1.8 (within ±0.05)

**Why:** $U = \tfrac{1}{2}LI^2 = 0.5 \times 0.40 \times 9.0 = 1.8$ J, the exact magnetic counterpart of $\tfrac{1}{2}CV^2$ for a capacitor.

Page: https://tryals.app/practice/physics-ii/electromagnetic-induction/an-inductor-of-0-40-h-carries-a-current-of-3-0-a-compute-the-stored

### 4. Which changes will induce an EMF in a stationary conducting loop?

A. The loop is being squeezed to a smaller area
B. The loop is rotating in a steady field
C. The magnetic field through it is increasing
D. A steady field passes through it unchanged

**Answer:** A. The loop is being squeezed to a smaller area; B. The loop is rotating in a steady field; C. The magnetic field through it is increasing

**Why:** Flux is $BA\cos\theta$, so changing $B$, $A$ or $\theta$ all induce an EMF. A completely steady situation changes no flux and induces nothing, however strong the field.

Page: https://tryals.app/practice/physics-ii/electromagnetic-induction/which-changes-will-induce-an-emf-in-a-stationary-conducting-loop

### 5. A generator coil has 200 turns, area 0.05 m$^2$, and spins at 100 rad/s in a field of 0.20 T. Set the peak EMF, in volts.

**Answer:** 200 (within ±20)

**Why:** $\mathcal{E}_{peak} = NBA\omega = 200 \times 0.20 \times 0.05 \times 100 = 200$ V. The output is sinusoidal and its peak scales with all four factors, which is why generators are wound with many turns and spun fast.

Page: https://tryals.app/practice/physics-ii/electromagnetic-induction/a-generator-coil-has-200-turns-area-0-05-m-and-spins-at-100-rad-s

### 6. Lenz's law dictates that an induced current opposes the change in flux rather than the applied magnetic field itself. What practical consequence follows when the external flux threading a circuit begins to decrease?

A. The induced field acts to cancel out the remaining field
B. The induced field reinforces the diminishing applied field
C. The induced EMF vanishes because the rate of change is negative
D. The induced current reverses continuously to resist decay

**Answer:** B. The induced field reinforces the diminishing applied field

**Why:** Opposing a decrease means attempting to maintain the original flux, so the induced magnetic field must point in the same direction as the external field. Students often confuse resisting a change with simply opposing the ambient field direction.

Page: https://tryals.app/practice/physics-ii/electromagnetic-induction/lenzs-law-dictates-that-an-induced-current-opposes-the-change-in

### 7. An electric field induced by a changing magnetic flux can be described by a scalar potential.

**Answer:** False

**Why:** False, the induced field has $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t \neq 0$, and only a curl-free field can be written as a gradient. This is exactly where the electrostatic potential picture stops working.

Page: https://tryals.app/practice/physics-ii/electromagnetic-induction/an-electric-field-induced-by-a-changing-magnetic-flux-can-be