Q.Use Lenz's law to determine the direction of induced current in the situations described by Fig. 6.16:

The key idea is Lenz’s law: the induced current opposes the change in magnetic flux through the loop.

(a) Irregular shape → circle

• The magnetic field is into the page ($\otimes$). As the loop becomes circular, its area increases, so the flux into the page increases.
• To oppose this increase, the induced current must produce its own magnetic field out of the page ($\odot$).
• By the right-hand rule, a current that gives an out-of-page field inside the loop must flow anticlockwise (as seen from above).

(b) Circular loop → narrow straight wire

• The magnetic field is out of the page ($\odot$). As the loop is squeezed into a narrow line, its area decreases, so the flux out of the page decreases.
• To oppose this decrease, the induced current must produce its own magnetic field out of the page ($\odot$) — i.e., it tries to maintain the original flux.
• By the right-hand rule, a current that gives an out-of-page field inside the loop must flow anticlockwise (as seen from above).

The Core Idea

Lenz’s law is the most elegant way to find induced current direction without getting tangled in right-hand rules. It states: the induced current flows in such a direction that its magnetic field opposes the change in the original magnetic flux through the loop. Think of it as the loop “fighting back” against whatever is happening to it — if flux is increasing, the loop tries to decrease it; if flux is decreasing, the loop tries to increase it.

In both parts of this problem, the magnetic field is uniform and constant. The only thing changing is the area of the loop. So the flux change comes purely from area change, and the induced current’s job is to oppose that area-driven flux change.

(a) Irregular wire turning into a circular shape

1. Identify the initial and final flux.

The field is uniform and directed into the page (shown by × marks). The wire starts as an irregular closed loop and is reshaped into a circle. A circle encloses the maximum area for a given perimeter, so the area increases as the shape becomes circular. Since $B$ is constant and into the page, the magnetic flux $\Phi = B A$ also increases.

2. Apply Lenz’s law.

Flux into the page is increasing. The induced current must create its own magnetic field that opposes this increase — that is, it must produce flux out of the page (to cancel some of the incoming flux).

3. Determine the current direction.

To produce a magnetic field out of the page, the induced current must flow anticlockwise (as seen from above). Use the right-hand grip rule: curl your fingers in the direction of current; your thumb points in the direction of the magnetic field inside the loop. Anticlockwise current gives a field out of the page.

(b) Circular loop being deformed into a narrow straight wire

1. Identify the initial and final flux.

Here the field is uniform and directed out of the page (shown by • dots). The loop starts as a circle and is squeezed into a narrow straight wire. As it narrows, the enclosed area decreases dramatically — eventually approaching zero. So the flux $\Phi = B A$ (out of the page) decreases.

2. Apply Lenz’s law.

Flux out of the page is decreasing. The induced current must oppose this decrease — that is, it must try to increase the out-of-page flux. So it must produce its own magnetic field out of the page.

3. Determine the current direction.

To produce a field out of the page, the induced current must again flow anticlockwise (same reasoning as part (a)).

Method: Lenz's Law When the Loop's Shape (Not Its Position) Changes

Some induction problems keep the loop in a fixed, uniform field but deform its shape — the induced current still comes from the resulting change in enclosed area, reasoned the same way as a loop physically leaving a field.

Steps

Step 1: Compare the enclosed area before and after the deformation

Reshaping a loop of fixed wire length (perimeter) changes its enclosed area because different shapes enclose different areas for the same perimeter — a circle encloses the maximum possible area for a given perimeter, so deforming towards a circle increases area, and squeezing a loop into a thin line drives the area towards zero.

Step 2: Translate the area change into a flux change

With $\vec B$ uniform and constant,

so the sign of $d\Phi_B/dt$ follows directly from whether $A$ is growing or shrinking.

Step 3: Apply Lenz's law to the flux trend

• Area (and hence flux) increasing → induced current opposes the increase → induced field points opposite to $\vec B$.
• Area (and hence flux) decreasing → induced current opposes the decrease → induced field points the same way as $\vec B$.

Step 4: Convert the required induced-field direction into a current direction

Use the right-hand grip rule: point your thumb along the induced field direction found in Step 3, and your curled fingers give the sense of the induced current around the loop.

Note the key insight: the shape of the loop (circle, irregular polygon, narrow line) never affects the direction rule itself — only whether the enclosed area, and hence the flux, is growing or shrinking matters. Two deformations that change flux for opposite reasons (one increasing into-page flux, one decreasing out-of-page flux) can still give the same current direction, because in both cases the induced current is trying to preserve the original flux value.