Problem#1
A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. 1. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) B is increasing; (b) B is decreasing; (c) B is constant with value B0. Explain your reasoning. |
| Fig.1 |
Answer:
(a) The field is into the page and is increasing so the flux is increasing. The field of the induced current is out of the page. To produce field out of the page the induced current is counterclockwise.
(b) The field is into the page and is decreasing so the flux is decreasing. The field of the induced current is into the page. To produce field into the page the induced current is clockwose.
(c) The field is constant so the flux is constant and there is no induced emf and no induced current.
Problem#2
The current in Fig. 2 obeys the equation I(t) = I0e-bt, where b > 0. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for t > 0.
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| Fig.2 |
Answer:
By Lenz’s law, the induced current flows to oppse the flux change that caused it.
The magnetic field is outward through the round coil and is decreasing, so the magnetic field due to the induced current must also point outward to oppse this decrease. Therefore the induced current is counterclockwise.
Careful! Lenz’s law does not say that the induced current flows to oppose the magnetic flux. Instead it says that the current flows to oppose the change in flux.
Problem#3
Using Lenz’s law, determine the direction of the current in resistor ab of Fig. 3 when (a) switch S is opened after having been closed for several minutes; (b) coil B is brought closer to coil A with the switch closed; (c) the resistance of R is decreased while the switch remains closed.
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| Fig.3 |
Answer:
Apply Lenz’s law, in the form that states that the lux of the induced current tends to opposite the change in flux.
(a) With the switch closed the magnetic filed of coil A is to the right at the location of coil B. When the switch is opened the magnetic field of coil A goes away. Hence by Lenz’s law the field of the current induced in coil B is to the right, to oppose the decrease in the flux in this direction. To produce magnetic field that is to the right the current in the circuit with coil B must flow through the resistor in the direction a to b.
(b) With the switch closed the magnetic field of coil A is to the right at the location of coil B. This field is stronger at points closer to coil A so when coil B is brought closer the flux through coil B increases. By Lenz’s law the field of the induced current in coil B is to the left, to oppose the increase in flux to the right. To produce magnetic field that is to the left the current in the circuit with coil B must flow through the resistor in the direction b to a.
(c) With the switch closed the magnetic field of coil A is to the right at the location of coil B. The current in the circuit that includes coil A increases when R is decreased and the magnetic field of coil A increases when the current through the coil increases. By Lenz’s law the field of the induced current in coil B is to the left, to oppose the increase in flux to the right. To produce magnetic field that is to the left the current in the circuit with coil B must flow through the resistor in the direction b to a.
Problem#4
A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in Fig. 4. Terminals a and b of winding A may be connected to a battery through a reversing switch. State whether the induced current in the resistor R is from left to right or from right to left in the following circumstances: (a) the current in winding A is from a to b and is increasing; (b) the current in winding A is from b to a and is decreasing; (c) the current in winding A is from b to a and is increasing.
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| Fig.4 |
Answer:
Apply Lenz’s law, the field of the induced current is directed to oppose the change in flux in the primary circuit.
(a) The magnetic field in A is to the left and is increasing. The flux is increasing so the field due to the induced current in B is to the right. To produce magnetic field to the right, the induced current flows through R from right to left.
(b) The magnetic field in A is to the right and is decreasing. The flux is decreasing so the field due to the induced current in B is to the right. To produce magnetic field to the right the induced current flows through R from right to left.
(c) The magnetic field in A is to the right and is increasing. The flux is increasing so the field due to the induced current in B is to the left. To produce magnetic field to the left the induced current flows through R from left to right whether the flux due to this field is increasing or decreasing.
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