(a) By how many percent is the torque of a motor decreased if its permanent magnets lose 5.0% of their strength? (b) How many percent would the current need to be increased to return the torque to original values?

Torque is a measurement of the force required to rotate an object around an axis. In linear kinematics, the force causes an object to accelerate; in angular kinematics, torque causes an object to accelerate. The torque can be estimated using the expression,

where the number of turns, $\mathbf{N}$, the current, $\mathbf{I}$, the area of the loop, $\mathbf{A}$, and the inclination angle of the loop, $\mathit{\theta }$.

The magnetic field strength in the initial $\left({\mathrm{B}}_{\mathrm{Initial}}\right)$and final $\left({\mathrm{B}}_{\mathrm{Final}}\right)$conditions can be calculated as,

$$\begin{gathered} {\mathrm{B}}_{\mathrm{Final}}={\mathrm{B}}_{\mathrm{Initial}}-5\%{\mathrm{B}}_{\mathrm{Initial}} \\ =0.95{\mathrm{B}}_{\mathrm{Initial}} \end{gathered}$$

We will use equation (1) for the torque and compare the initial and the final torque, and respectively.

Now, let us calculate the torque using the above calculation for magnetic field strength, we will get,

To get the decrease in the torque, we will subtract the final torque from the initial torque.

So, the decrease in the torque is${}^{\mathbf{5}\mathbf{\%}}$.

In this case, the value of the initial torque is the same as the final torque. We will compare the two values together to get the increase in the current.

Let us calculate the current,

Rearranging the above equation, we will get

$$\begin{gathered} \frac{{\mathrm{I}}_{\mathrm{Final}}}{{\mathrm{I}}_{\mathrm{Initial}}}=\frac{{\mathrm{B}}_{\mathrm{Initial}}}{{\mathrm{B}}_{\mathrm{Final}}} \\ =\frac{100}{95} \\ =1.053 \\ \mathrm{Further},\mathrm{we}\mathrm{will}\mathrm{get}, \end{gathered}$$

$$\begin{gathered} {\mathrm{I}}_{\mathrm{Final}}=1.053{\mathrm{I}}_{\mathrm{Initial}} \\ ={\mathrm{I}}_{\mathrm{Initial}}+0.053{\mathrm{I}}_{\mathrm{Initial}} \\ ={\mathrm{I}}_{\mathrm{Initial}}+5.3\%{\mathrm{I}}_{\mathrm{Initial}} \end{gathered}$$

Hence, the required percent of the increase in the current is 5.3%.