Question

Using Lenz’s law, determine the direction of the current in resistor ab of Fig. E29.19 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.

Short answer

a) The direction of current is from a to b.

b) The direction of current is from b to a.

c) The direction of current is from b to a.

Step-by-step solution

About Lenz law.

Lenz's law: Lenz's law states that an induced current or emf always tends to oppose or cancel out the change that caused it.

Right-hand rule for a magnetic field produced by a current in a loop: When the fingers of your right hand curl in the direction of the current, your right thumb points in the direction of the magnetic field lines.

Right-hand thumb rule.

When the switch S is closed, the current flows in the circuit in the direction from the +terminal to the terminal of the battery.

Using the right-hand rule on loop A, we curl the fingers of the right hand, and we find

that the right thumb is pointing to the right

Thus, whenever the switch S is closed, the magnetic field produced by the current in

Thumb rule.

(a) When switch S is opened after having been closed for several minutes, the current of the coil is reduced to zero, and so is the magnetic field.

So the magnetic field at coil B is decreasing to zero.

Applying Lena's law, this change in the magnetic field at coil B induces a current in

coil B to oppose this change.

So, the magnetic field produced by the induced current is to increase the magnetic

field to the right

Thus, the magnetic field produced by coil B is to the right

Using the right-hand rule, point your right thumb in the direction of the magnetic field

to the right, so your right fingers now curl in the direction of the induced current.

Following the direction of the induced current, we find it flows through the resistor ab

from a to b.

Thumb rule.

(b) When coil B to brought closer to coil A, the magnetic field at coil B increases.

Applying Lenz's law, an induced current is produced in coil B to oppose this

change.

So, the magnetic field produced by the induced current is to decrease the magnetic

field.

Thus, the magnetic field produced by loop B is to the left (to oppose the direction of

the magnetic field of coil A).

Using the right-hand rule, point your right thumb in the direction of the magnetic field

to the left, so your right fingers now curl in the direction of the induced current in coil

B.

Following the direction of the induced current, we find it flows through the resistor ab

from b to a

Lenz law and Ohms law.

(c) When the resistance of R is decreased while the switch remains closed, and by

applying Ohm's law, while the potential difference is constant, the current in the circuit increases.

$$\begin{gathered} V=IR \\ I=\frac{V}{R} \end{gathered}$$

The change in the current flows in coil A results in a change in the magnetic field

produced by the coil, according to the equation

$B=\frac{{\mu }_{0}I{a}^2}{2(a^2+x^2{)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

Since the current in coil A is increasing, the magnetic field at coil B is also increasing

the right direction.

Applying Lenz's law, an induced current is produced in coil B to oppose this

change.

So, the magnetic field produced by the induced current is to decrease the Magne

the field in the right direction.

Thus, the magnetic field produced by loop B is to the left (to oppose the direction of

the magnetic field of coil A).

Using the right-hand rule, point your right thumb in the direction of the magnetic field

to the left, so your right fingers now curl in the direction of the induced current in coil B.

Following the direction of the induced current, we find it flows through the resistor ab

from b to a.