Question

The wire loop in Fig. 30-22ais subjected, in turn, to six uniform magnetic fields, each directed parallel to the axis, which is directed out of the plane of the figure. Figure 30- 22bgives the z components Bz of the fields versus time . (Plots 1 and 3 are parallel; so are plots 4 and 6. Plots 2 and 5 are parallel to the time axis.) Rank the six plots according to the emf induced in the loop, greatest clockwise emf first, greatest counter-clockwise emf last.

Short answer

The rank of six plots according to the emf induced in the loop is plot 1 and 3 ties, plot 2 and 5 ties, and plot 4 and 6 ties.

Step-by-step solution

Step 1: Given

The wire loop is subjected to six uniform magnetic fields, parallel to Z, directed out of the page.

Determining the concept

The direction of the induced emf is the same as that of the induced current. The direction of the induced current is such that its magnetic field opposes the change in the applied magnetic field.

Formulae are as follows:

$\epsilon =\frac{-d\varphi }{dt}$,

$\varphi =B.A$,

where,

$\epsilon$= Induced emf,

$\varphi$= change in magnetic flux,

B = magnetic field,

A = surface area.

Determining the rank of six plots according to the emf induced in the loop

According to Faraday’s law, the induced emf depends upon the changing magnetic flux. i.e.

$\epsilon =\frac{-d\varphi }{dt}$,

The magnitude of the flux is determined by the magnetic field.

$\varphi =B.A$,

Thus, the magnitude of the emf depends upon the changing applied to the magnetic field.

$\epsilon =\frac{-d\varphi }{dt}=-A\frac{dB}{dt}$,

In the case of plots 1 and 3, the plot has a positive slope. So, the magnetic field is increasing with time and is directed out of the page. Thus, the induced magnetic field will be opposite to it i.e. it will be directed into the page. Using the right-hand rule, the direction of the induced current will be clockwise. The direction of the emf is the same as that of the induced current hence, it will also be clockwise. The magnitude of the emf will be the same for both, since the magnitude of the flux change is the same.

In the case of plots 2 and 4, the magnetic field remains constant with time. Hence, no change inthe flux occurs. Thus, the induced emf will be zero for these plots.

In the case of plots 4 and 6, the plots have negative slopes. Thus, the magnetic field is decreasing with time. Thus, the induced field will try to oppose the decrease. Thus, the direction of the induced field will be the same as that of the applied field. So, it will be directed out of the page. Thus, the right-hand rule says, the direction of the induced current will be counter-clockwise. This is the same direction for the induced emf. The magnitude of the emf will be the same for both, since the magnitude of the flux change is the same.

Hence,the rank of six plots according to the emf induced in the loop is plot 1 and 3 ties, plot 2 and 5 ties, and plot 4 and 6 ties.

The magnitude of theinduced emf is given by Faraday’s law while Lenz’s law helps determine its direction. Thus,the magnitude and the direction of the induced emf using the information given by the plots can be determined.