VerifiedHint: To find the magnitude of the torque ($\tau$) acting on an electric dipole in an electric field when it is at an angle with the field, you can use the formula:
$\tau = p \cdot E \cdot \sin(\theta)$
Where:
$\tau$ is the torque.
$p$ is the dipole moment $4 \times 10^{-9} \text{Cm}$.
$E$ is the electric field magnitude $5 \times 10^{4} \text{N/C}$.
$\theta$ is the angle between the dipole moment and the electric field $30^\circ$.
Step-by-Step Solution:
Let's calculate the magnitude of the torque ($\tau$) acting on the dipole using the formula provided in the hint:
$\tau = p \cdot E \cdot \sin(\theta)$
Substitute the given values:
$p = 4 \times 10^{-9} \text{Cm}$
$E = 5 \times 10^{4} \text{N/C}$
$\theta = 30^\circ$ (Make sure to convert to radians for the trigonometric function.)
$\theta = 30^\circ \times \left(\dfrac{\pi}{180}\right) = \dfrac{\pi}{6} \text{ radians}$
Now, calculate $\tau$:
$\tau = (4 \times 10^{-9} \text{Cm}) \cdot (5 \times 10^{4} \text{N/C}) \cdot \sin\left(\dfrac{\pi}{6}\right)$
Calculate the sin value:
$\sin\left(\dfrac{\pi}{6}\right) = \dfrac{1}{2}$
Now, calculate $\tau$:
$\tau = (4 \times 10^{-9} \text{Cm}) \cdot (5 \times 10^{4} \text{N/C}) \cdot \dfrac{1}{2}$
Calculate the torque $\tau$:
$\tau = 10^{-4} \text{Nm}$
So, the magnitude of the torque acting on the dipole is $10^{-4} \text{Nm}$.
Note: The torque acts to align the dipole with the electric field, and its magnitude is directly proportional to the dipole moment, the electric field strength, and the sine of the angle between them. In this case, the torque is $10^{-4} \text{Nm}$.
