Q.Figure 6.7 shows planar loops of different shapes moving out of or into a region of a magnetic field which is directed normal to the plane of the loop away from the reader. Determine the direction of induced current in each loop using Lenz's law.

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

Step 1 – Identify flux change for each loop

The magnetic field is uniform and directed into the page (× symbols).

• Loop (i) is entering the field → flux into the page is increasing.
• Loop (ii) is leaving the field → flux into the page is decreasing.
• Loop (iii) is moving out of the field → flux into the page is decreasing.

Step 2 – Apply Lenz’s law to find induced field direction

• For increasing flux (i): induced field must oppose the increase → point out of the page (⊙).
• For decreasing flux (ii, iii): induced field must support the decreasing flux → point into the page (×).

Step 3 – Use right-hand rule to get current direction

Curl fingers in direction of induced current; thumb gives induced field direction.

• (i) Induced field ⊙ → current flows anticlockwise (abcd).
• (ii) Induced field × → current flows clockwise (abc).
• (iii) Induced field × → current flows clockwise (abcd).

The core idea

Lenz’s law is a statement of energy conservation: the induced current creates a magnetic field that opposes the change in the original flux through the loop. If flux is increasing, the induced field points opposite to the original field; if flux is decreasing, the induced field points in the same direction as the original field. The direction of current is then given by the right-hand rule (curl fingers along current, thumb gives field direction).

The magnetic field in all three cases is uniform and directed into the page (the × symbols). So the original field $\vec{B}$ points away from the reader, into the plane.

Step-by-step for each loop

1. Loop (i) — rectangle entering the field

The loop moves down-right, so more of its area enters the region of ×’s. The flux through the loop (into the page) is increasing.

To oppose this increase, the induced magnetic field must point out of the page (opposite to the original field). Using the right-hand rule: if the thumb points out of the page, the curled fingers show the current direction — anticlockwise (counterclockwise) as seen from above.

Result for (i): Induced current flows anticlockwise (a → b → c → d → a).

2. Loop (ii) — triangle leaving the field

The triangle moves to the right, so its area inside the ×-region is decreasing. The flux into the page is decreasing.

To oppose this decrease, the induced field must point into the page (same direction as the original field). Right-hand rule: thumb into the page → fingers curl clockwise.

Result for (ii): Induced current flows clockwise (a → c → b → a, or whichever order follows the vertices).

3. Loop (iii) — irregular loop moving out

The loop moves down-left, so its area inside the ×-region is decreasing. The flux into the page is decreasing.

Same logic as (ii): decreasing flux → induced field must point into the page → current is clockwise.

Result for (iii): Induced current flows clockwise (a → b → c → d → a).

Summary table

Method: Lenz's Law for Loops Moving Through a Field Boundary

This method finds the direction of induced current whenever a closed loop's area inside a field region changes — the loop entering, leaving, or moving across a field boundary.

Steps

Step 1: Note the field's direction and mark it on the loop

Identify whether $\vec B$ points into or out of the page (× or • symbols). This is your reference direction for everything that follows.

Step 2: Decide whether the enclosed flux is increasing or decreasing

Ask: as the loop moves, is more of its area now inside the field region (flux increasing) or less (flux decreasing)? A loop entering a field region has increasing flux; a loop leaving has decreasing flux — regardless of its shape.

Step 3: Apply Lenz's law to get the induced field's direction

• If flux is increasing, the induced current must create a field that opposes it — i.e., points opposite to $\vec B$.
• If flux is decreasing, the induced current must create a field that supports it — i.e., points the same way as $\vec B$.

Step 4: Use the right-hand (grip) rule to convert field direction into current direction

Point your thumb along the induced-field direction found in Step 3; your curled fingers show the sense of circulation of the induced current around the loop. Read off the current direction using the loop's labelled corners.

Applying this same four-step process to each panel of a multi-loop figure — independently, since each loop's own entering/leaving motion decides its own answer — is exactly how every "determine the direction of induced current" question of this type is solved.