Question

Does the magnitude of the net force on the current loop of Figs. 32-12a and b increase, decrease, or remain the same if we increase (a) the magnitude of ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$and (b) the divergence of ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$?

Short answer

1. The magnitude of the net force on the current loop of figures 32-12a and b remains the same if the magnitude of ${\stackrel{\rightharpoonup }{B}}_{ext}$is increased.
2. The magnitude of the net force on the current loop of figures 32-12a and b remains the same if the divergence of role="math" localid="1663045183942" width="30" height="28">B⇀extis increased.

Step-by-step solution

Determining the concept

From the formula for net force acting on the current loop in the non-uniform magnetic field, we can find whether the magnitude of the net force on the current loop increases, decreases, or remains the same, if the magnitude of ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$and divergence of ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$is increased.

(a) Determining whether the magnitude of the net force on the current loop increases, decreases, or remains the same.

The net force on the current loop in figures 32-12a and b will be upwards because if the loop is divided into small elements and they are split into horizontal and vertical components, the horizontal component of force due to one element is canceled bytheopposite side element. Therefore, the net force is due to vertical components only.

If we increase the magnetic field, the force on the loop will increase, and hence the length of the vertical component will also increase.

Hence, the increase in the magnitude of ${\stackrel{\rightharpoonup }{B}}_{ext}$will increase the net force on the current loop.

Therefore, the magnitude of the net force on the current loop of figures 32-12a and b will increase if the magnitude of role="math" localid="1663045371493" width="30" height="28">B⇀extis increased.

(b) Determining whether the magnitude of the net force on the current loop increases, decreases, or remains the same if the divergence of B⇀ext is increased.

Since the magnitude of ${\stackrel{\rightharpoonup }{B}}_{ext}$increases the net force on the current loop will also increase. As the net force is increased with an increase in ${\stackrel{\rightharpoonup }{B}}_{ext}$, the increase in divergence of ${\stackrel{\rightharpoonup }{B}}_{ext}$will also increase the net force on the current loop.

Therefore, the magnitude of the net force on the current loop of figures 32-12a and b increases if the divergence of ${\stackrel{\rightharpoonup }{B}}_{ext}$is increased.