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Question:

An electric dipole with a dipole moment of $4 \times 10^{-9}  \text{Cm}$ is aligned at $30^\circ$ with the direction of a uniform electric field of magnitude $5 \times 10^{4}  \text{N/C}$. Calculate the magnitude of the torque acting on the dipole.

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Last updated date: 18th Jun 2024
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Answer
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Hint: 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 $10^{-4}  \text{Nm}$.