Two charges each of magnitude 0.01 C and separated by a distance of 0.4 mm constitute an electric dipole. If the dipole is placed in an uniform electric field of 10 dyne/C making 30º angle with $\overrightarrow E$, the magnitude of torque acting on dipole is:

Question

A thin half ring of radius $35 \text{ cm}$ is uniformly charged with a total charge of $Q$ coulomb. If the magnitude of the electric field at centre of the half ring is $100 \text{ V/m}$, then the value of $Q$ is ______ $\text{nC}$.
($\epsilon_o = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2$ and $\pi = 3.14$)

Answer 1

To determine the torque acting on the electric dipole, we will employ the formula for the torque $ \tau $ on a dipole in a uniform electric field:

$\tau = pE \sin\theta$

where:

$ p $ is the dipole moment.
$ E $ is the electric field strength.
$ \theta $ is the angle between the dipole moment and the electric field.

Step 1: Calculate the Dipole Moment $(p)$

The dipole moment is given by:

$ p = q \cdot d $

where:

$ q = 0.01 \text{ C} $ (charge on each charge of the dipole)
$ d = 0.4 \text{ mm} = 0.4 \times 10^{-3} \text{ m} $ (separation between the charges)

Substitute the values:

$ p = 0.01 \times 0.4 \times 10^{-3} = 4 \times 10^{-6} \text{ Cm}$

Step 2: Convert Electric Field from dyne/C to N/C

Given:

$ E = 10 \text{ dyne/C} $

We know:

$ 1 \text{ dyne} = 10^{-5} \text{ N} $

Therefore:

$ E = 10 \times 10^{-5} \text{ N/C} = 10^{-4} \text{ N/C} $

Step 3: Calculate the Torque $(\tau)$

Use the dipole torque formula:

$ \tau = pE \sin\theta $

Substitute the known values:

$ p = 4 \times 10^{-6} \text{ Cm} $
$ E = 10^{-4} \text{ N/C} $
$ \theta = 30^\circ $
$ \sin 30^\circ = \frac{1}{2} $

Thus, we have:

$ \tau = 4 \times 10^{-6} \times 10^{-4} \times \frac{1}{2} $

$ \tau = 2 \times 10^{-10} \text{ Nm} $

Hence, the magnitude of the torque acting on the dipole is $2.0 \times 10^{-10} \text{ Nm}$, which matches with the correct option.

Answer 2

To determine the torque acting on the electric dipole, we will employ the formula for the torque $ \tau $ on a dipole in a uniform electric field:

$\tau = pE \sin\theta$

where:

$ p $ is the dipole moment.
$ E $ is the electric field strength.
$ \theta $ is the angle between the dipole moment and the electric field.

Step 1: Calculate the Dipole Moment $(p)$

The dipole moment is given by:

$ p = q \cdot d $

where:

$ q = 0.01 \text{ C} $ (charge on each charge of the dipole)
$ d = 0.4 \text{ mm} = 0.4 \times 10^{-3} \text{ m} $ (separation between the charges)

Substitute the values:

$ p = 0.01 \times 0.4 \times 10^{-3} = 4 \times 10^{-6} \text{ Cm}$

Step 2: Convert Electric Field from dyne/C to N/C

Given:

$ E = 10 \text{ dyne/C} $

We know:

$ 1 \text{ dyne} = 10^{-5} \text{ N} $

Therefore:

$ E = 10 \times 10^{-5} \text{ N/C} = 10^{-4} \text{ N/C} $

Step 3: Calculate the Torque $(\tau)$

Use the dipole torque formula:

$ \tau = pE \sin\theta $

Substitute the known values:

$ p = 4 \times 10^{-6} \text{ Cm} $
$ E = 10^{-4} \text{ N/C} $
$ \theta = 30^\circ $
$ \sin 30^\circ = \frac{1}{2} $

Thus, we have:

$ \tau = 4 \times 10^{-6} \times 10^{-4} \times \frac{1}{2} $

$ \tau = 2 \times 10^{-10} \text{ Nm} $

Hence, the magnitude of the torque acting on the dipole is $2.0 \times 10^{-10} \text{ Nm}$, which matches with the correct option.

Two charges each of magnitude 0.01 C and separated by a distance of 0.4 mm constitute an electric dipole. If the dipole is placed in an uniform electric field of 10 dyne/C making 30º angle with \(\overrightarrow E\), the magnitude of torque acting on dipole is:

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The Correct Option is B

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