Question

In Fig. 30-23, a long straight wire with current ipasses (without touching) three rectangular wire loops with edge lengths L, 1.5L, and 2L. The loops are widely spaced (so as not to affect one another). Loops 1 and 3 are symmetric about the long wire. Rank the loops according to the size of the current induced in them if current iis (a) constant and (b) increasing, greatest first.

Short answer

a) The rank of loops according to the size of the induced current is: all are the same (zero).

b) The rank of loops according to the size of the induced current is: loop 2, loop 1, and 3tie (zero).

Step-by-step solution

Given

a) The 3 wire loops have current-carrying wire passing through them.

b) The 3 loops have edge lengths L, 1.5 L, and 2L.

c) Loops 1 and 3 are symmetric about the current-carrying wire.

Determining the concept

The changing current produces the changing magnetic field. This field in turn produces induced current in the loops. The direction of the induced current is given by Lenz’s law.

Formulae are as follows:

$E=-N\frac{d\varnothing }{dt}$,

Where,

E = induced emf,

$d\varnothing$= change in magnetic flux,

N = number of turns in coil,

dt = change in time.

(a) Determining the rank of loops according to the size of the induced current if the current is constant

According to Faraday’s law, the current is induced in a loop if the magnetic flux through it changes. But in case (a), the current i remains constant. Hence, it will not produce any change in the magnetic field (also the flux). So, no current is induced in all three loops.

Hence, the rank of loops according to the size of the induced current is are all same (zero).

(b) determining the rank of loops according to the size of induced current if the current is increasing

As the current increases in case (b), it will change the magnetic flux through all three loops. For loops 1 and 3, the current-carrying wire is placed symmetrically. Thus, the direction and the magnitude of the magnetic field in the upper half of the loop are exactly opposite to that in the other half of the loop. Thus, the direction and magnitude of the induced current are also opposite in each half of the loop. This gives the net induced current in each of loops 1 and 3 to be zero.

For loop 2, the wire is not placed symmetrically. So, the magnetic flux in the upper half of the loop is more than that in the lower half of the loop. Also, their directions are exactly opposite. Hence, the induced current in the upper half of the loop will be more than that in the lower half which is also in opposite directions. In the upper half of the loop, the direction of the magnetic field is out of the page. Thus, the loop has the net-induced current in the clockwise direction. Hence, it is ranked first.

Hence, the rank of loops according to the size of the induced current of loop 2, loop 1, and 3 tie (zero).

The current-carrying wire will have its magnetic field. When the magnitude of the current changes, it will change the magnetic field and the flux. This will induce the current in the loop. Thus, the magnitude of the induced current depending upon the magnitude of the magnetic flux through each loop can be determined.