Use Lenz’s law with a clear viewing direction

Oppose the change, not necessarily the existing field

Lenz’s law describes the direction of induction: the induced current’s magnetic effect opposes the change in applied flux that drives it. The word “change” matters. A decreasing positive flux calls for a different response from an increasing positive flux, even though the applied field points the same way in both cases.

Keep the normal along +z (0°) first. Positive circulation is counterclockwise viewed from the normal tip looking toward the coil. A positive current has a magnetic effect through the coil along the normal; a negative current has the opposite effect. This is a direction prediction. The browser does not calculate the magnitude of the coil’s self-generated field or its feedback on the prescribed source.

Apply a four-step direction check

1. State the normal and the viewing side. At 0°, the normal is +z and the viewer is on the +z side looking toward the coil.
2. Determine whether the signed applied flux is increasing or decreasing. Use the graph’s slope, not merely whether the field is positive or negative.
3. Choose the induced magnetic effect that opposes that change in flux.
4. Use the current right-hand rule to infer circulation, then check the sign of emf and current in the ledger.

For s = +1 T/s, positive flux increases. The induced effect through the coil must point toward −z. That requires clockwise current viewed from +z, which is negative in this reference. The ledger gives −0.400 V and −40 mA. The minus sign in Faraday’s law and the direction argument agree.

For s = −0.5 T/s, the applied field falls from +1 T to zero in two seconds. The positive flux decreases, so the induced effect points toward +z, tending to sustain it. Current is counterclockwise from +z: +20 mA. Here the induced effect points the same way as the applied field during most of the interval. Saying it always opposes the existing field would give the wrong direction.

Capture these two ramps and explain the difference without relying on the current sign alone.

Follow a field reversal without changing the rule

At s = −1 T/s, the field crosses zero after one second and becomes negative. Before that instant, positive flux is decreasing; afterward, negative flux is becoming more negative. In both intervals the signed flux slope is negative. The induced effect points toward +z throughout, so the current remains counterclockwise from +z and the emf remains +0.400 V. It does not reverse at the field’s zero crossing.

At zero ramp rate, no applied flux change drives an emf. At 90°, the field changes within the coil plane but its normal component and flux remain zero, giving no net loop emf in this ideal geometry. Zero net loop emf is the conclusion; it does not require the electric field to be zero at every point in space.

Keep the viewing reference consistent

Compare 0° and 180° at the same ramp rate. These have the same unoriented plane and opposite normal references. Flux, emf and signed current reverse together. At +1 T/s, the 0° case says negative current, clockwise viewed from +z. The 180° case says positive current, counterclockwise viewed from −z. These describe the same physical circulation when viewed from one fixed side. Reversing a reference is not a new energy source or a contradiction of Lenz’s law.

At an intermediate angle such as 60°, use the component of the induced magnetic effect along the normal to judge whether it opposes flux change. It need not be exactly opposite the entire applied-field vector. The source field is prescribed and the self-field is neglected in the browser’s calculation; the qualitative direction check does not turn it into a coupled electromagnetic simulation.

Account for energy without claiming a free generator

The default case dissipates 32 mJ in the resistance over the two-second interval. That energy does not come from the definition of a minus sign. An external system that creates and changes the applied field must participate in the energy transfer. A changing magnetic field produces an electric field that can do work on charges; this is consistent with the previous result that the magnetic part of the force alone does no work on an individual moving charge.

The browser prescribes Bz(t), so it cannot determine the total input energy, source efficiency, stored field-energy change or forces on the source apparatus. Lenz’s law supplies the opposing direction, while a complete source-and-coil model would be needed for a complete energy budget. Do not equate the displayed resistor heat with the total energy cost of creating the applied field.

Transfer the direction reasoning

A coil normal points toward you. An applied field pointing away from you is getting weaker in magnitude. Determine the sign of flux, its slope, the induced magnetic effect and the current direction viewed from your side. Then consider that same away-pointing field getting stronger.

Check the Lenz transfer

Away-pointing field has negative flux in this reference. As it weakens, that flux becomes less negative, so the slope is positive. Induced emf and current are negative: clockwise viewed from your side, producing an away-pointing effect that tends to sustain the decreasing magnitude. If the away-pointing field strengthens, flux becomes more negative, the slope is negative, and current is counterclockwise with an effect toward you. The sign of the field alone cannot distinguish these cases.

Give a direction argument, a consistent view and a limitation in your reflection. Public practice answers check particular claims; they do not establish independent apparatus skill or an experimentally verified energy budget.

Reference: OpenStax: Lenz’s law. The signed-ramp comparisons, reference checks and transfer scenario are original.