Question

(a) At what angle θ is the torque on a current loop 90.0% of maximum? (b) 50.0% of maximum? (c) 10.0% of maximum?

Short answer

The required angles are

(a) $64°$

(b) $30°$

(c) $5.7°$

Step-by-step solution

Definition of Torque.

Torque is a measurement of the force required to rotate an object around an axis and be expressed as,

$\mathbf{\tau }\mathbf{=}\mathbf{NIABsin}\mathbf{\left(}\mathbf{\theta }\mathbf{\right)}$………………(1)

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, $\mathbf{\theta }$.

Strategy

The maximum torque is obtained when$\mathrm{\theta }=90°$and therefore$\sin \left(90°\right)=1$, and in that case, the equation (1) can be written as,

${\tau }_{\mathrm{mas}}=\mathrm{NIAB}$…………….……(2)

we will compare the torque in each case with the maximum torque to get the angle.

$$\begin{gathered} \frac{\mathrm{\tau }}{{\mathrm{\tau }}_{\mathrm{mas}}}=\frac{\mathrm{NIAB}\sin \left(\mathrm{\theta }\right)}{\mathrm{NIAB}} \\ =\sin \left(\mathrm{\theta }\right) \end{gathered}$$ ……………………..(3)

Step 3:  Calculating the angle at torque is of the maximum torque (a)

Let us calculate the angle using equation (3), such that,

$$\begin{gathered} \sin \left(\mathrm{\theta }\right)=\frac{90}{100} \\ \mathrm{\theta }=\arcsin \left(\frac{90}{100}\right) \\ \mathrm{\theta }=64° \end{gathered}$$

Hence, the required value is $64°$.

 Calculating the angle at torque is  of the maximum torque(b)

Calculate the angle using equation (3), which will give,

$$\begin{gathered} \sin \left(\mathrm{\theta }\right)=\frac{50}{100} \\ \mathrm{\theta }=\arcsin \left(\frac{50}{100}\right) \\ \mathrm{\theta }=30° \end{gathered}$$

Hence, the required angle is $30°$.

Calculating the angle at torque is  of the maximum torque(c)

Considering the given information and using equation (3), we will get

$$\begin{gathered} \sin \left(\mathrm{\theta }\right)=\frac{10}{100} \\ \mathrm{\theta }=\arcsin \left(\frac{10}{100}\right) \\ \mathrm{\theta }=5.7° \end{gathered}$$

Therefore, the required angle is $5.7°$