Lenz's Law and Conservation of Energy

Answers & Solutions

Work the questions above first, then check yourself here. Each answer is set out the way it should be written in the paper.

1. 1.↑ back to the question

   As the N-pole of the magnet approaches the coil, by Lenz's law the induced current in the coil must oppose the increasing flux, so it flows in a direction that makes the coil face nearer to the magnet behave like a north pole (to repel the approaching magnet). Working out the resulting sense of current flow in the coil (clockwise, as viewed from the magnet's side) and tracing it around to the capacitor plates, plate A acquires positive polarity with respect to plate B. Polarity of plate A will be positive with respect to plate B in the capacitor, as the induced current is in the clockwise direction.

2. 2.↑ back to the question

   As the magnet is moved away from coil PQ, the flux linked with PQ decreases, and by Lenz's law the induced current in PQ opposes this decrease (tries to maintain the flux). As the same magnet approaches coil CD, the flux linked with CD increases, and the induced current in CD opposes this increase. According to Lenz's law, working through the geometry of the windings as drawn, the direction of the induced current in loop PQ is from P to Q, and in loop CD is from C to D.

3. 3.↑ back to the question

   Since ring 1 (above the wire) and ring 2 (below the wire) lie on opposite sides of the current-carrying wire, the magnetic field of the wire threads them in opposite senses (into the page for one, out of the page for the other), even though the current in the wire is increasing in the same sense for both. By Lenz's law, the induced current in each ring opposes the increasing flux threading it. The direction of the induced current in metal ring 1 is clockwise, and in metal ring 2 it is anticlockwise, when the current $I$ in the wire is increasing steadily.

4. 4.↑ back to the question

   As the current-carrying solenoid approaches the loop, the magnetic flux linked with the loop (due to the solenoid's field) increases. By Lenz's law, the induced current in the loop must oppose this increasing flux, i.e. it must set up a magnetic field that opposes the approaching solenoid's field within the loop. So the induced current in the loop circulates in the sense OPPOSITE to the current in the solenoid, viewed from either side along the common axis: the loop then presents a like pole to the approaching solenoid and repels it.

   [Note on this record: the original figure was lost in import, and with it the direction of the solenoid's own current. The handedness of the induced current follows from that direction, so the answer is stated relative to it — which is what the question is testing — rather than as a fixed clockwise or anticlockwise.]

5. 5.↑ back to the question

   The current in the straight wire (flowing from B to A) is decreasing, so the magnetic field it produces (and hence the flux linked with the loop placed above it) is decreasing. By Lenz's law, the induced current in the loop must oppose this decrease, i.e. it must flow so as to reinforce (sustain) the existing magnetic field in its original direction. With the current flowing from $B$ (right) to $A$ (left), the right-hand rule puts the wire's field INTO the page at the loop above it. To sustain that decreasing inward field, the induced current must itself produce an inward field inside the loop, so it flows CLOCKWISE.

   [Note on this record: the original figure was lost in import. The positions of $A$ and $B$ have been stated in the question, because the answer is a handedness and is meaningless without them. They are the positions the recorded answer already assumed; no answer has been changed.]

6. 6.↑ back to the question

   As the flexible wire straightens out into a circle, the area enclosed by it increases, so the flux (into the page) linked with it increases. By Lenz's law, the induced current must oppose this increase, i.e. it must produce a magnetic field out of the page inside the loop, which (by the right-hand rule) means the induced current flows anticlockwise as viewed by the reader. Physically, this also means the magnetic force on each current-carrying element of the wire acts outward, radially, which is exactly the force needed to expand the irregular wire into a circular shape. The induced current flows in the anticlockwise direction in the loop, i.e. in the sequence c – b – a – d.

7. 7.↑ back to the question

   (i) For a conducting disc of radius $r$ rotating with angular speed $\omega$ in a field $B$ perpendicular to its plane, the emf induced between the centre and the rim is $|\varepsilon| = \dfrac{B\omega r^2}{2}$ This expression depends only on $B$, $\omega$ and $r$ — it does not depend on the material (resistivity) of the disc. Since both discs are identical in size and are rotated with the same $\omega$ in the same field $B$, the induced emf is the SAME in both the copper and the aluminium disc. (ii) The induced current is $I = \dfrac{\varepsilon}{R}$, where $R$ is the resistance of the disc between its centre and edge. Copper has a lower resistivity (and hence lower resistance $R$) than aluminium. Since the emf is the same in both, the disc with the smaller resistance carries the larger current: the copper disc, having less resistance, will therefore have a larger induced current than the aluminium disc.

8. 8.↑ back to the question

   Lenz's law: The direction of the induced emf and the induced current in a circuit are always such that they oppose the cause (the change in magnetic flux) that produces them. Lenz's law is a consequence of the law of conservation of energy. Here, the North pole of the magnet is approaching the coil, so the induced current, as viewed from the side facing the magnet, flows in such a way that this face behaves like a north pole — so as to repel the approaching magnet. A south pole is therefore developed on the face of the coil away from the magnet. Tracing this current around the coil to the capacitor, the flow of induced current is clockwise; hence plate A acquires positive polarity and plate B acquires negative polarity.

9. 9.↑ back to the question

   Lenz's law states that the direction of the induced emf and the direction of the induced current are always such that they oppose the cause which produces them. Simple experiment/example: When the N-pole of a bar magnet is moved towards a coil connected to a galvanometer, the induced current in the coil flows in the anticlockwise direction (as viewed from the magnet's side), so as to make that face of the coil a north pole, which opposes the motion of the magnet towards the coil by exerting a repulsive force on it. Similarly, when the N-pole of the magnet is moved away from the coil, the induced current flows in the clockwise direction, developing a south pole on that face, which attracts the receding magnet, again opposing its motion (this time, opposing its departure). Justification via conservation of energy: In order to keep changing the magnetic flux linked with the circuit (i.e. to keep moving the magnet against the opposition offered by the induced emf), some work must be done by the external agent (or energy must be spent). This energy spent ultimately appears in the circuit as electrical energy. Suppose, for contradiction, that Lenz's law were not valid, i.e. suppose the induced current flowed in the direction opposite to that predicted by Lenz's law. Then the resulting force on the magnet would make it move faster and faster on its own — the magnet would gain kinetic energy without any equivalent expenditure of energy by an external agent. This would amount to a perpetual motion machine, which violates the law of conservation of energy. Hence Lenz's law must be valid, and it is indeed a direct consequence of the law of conservation of energy.