Question

An electric Dipole of length $10cm$ having charges $\pm 6 \times {10^{ - 3}}C$ placed at ${30^o}$ w.r.t a uniform electric field experiences a torque of magnitude $6\sqrt 3 Nm$. Calculate
(1) Magnitude of electric Field
(2) Potential energy of dipole.

Answer 1

At first, we need to calculate the Dipole moment of the dipole. From the obtained value we can get the value of potential energy. Using the equation of torque, the value of the electric field can be obtained.

Formula Used:
1. $p = q \cdot d$
Where,
$\vec p =$Dipole moment
$d =$Distance between the dipoles
$q =$Charge of the dipole
2. $\tau = \vec p \times \vec E = pE\sin \theta$
Where,
$\tau =$ Torque
Dipole Moment
$\vec E =$Electrical Field
3. $U = \vec p \cdot \vec E = pE\cos \theta$
$\vec p =$Dipole Moment
$\vec E =$Electric Field
$U =$Potential Energy

Complete step-By-Step Answer:
Given:
Distance between the dipoles $= 10cm$ denoted by $d$
Charge of the dipole$= \pm 6 \times {10^{ - 3}}C$ denoted by $q$
Therefore, dipole moment is given by:
$p = q \cdot d$
Where,
$\vec p =$Dipole moment
$d =$Distance between the dipoles
$q =$Charge of the dipole
$p = \pm 6 \times {10^{ - 3}}C \times {10^{ - 2}}$
Make sure, centimeter is converted to meters. So substituting the values of charge and distance we get,
$p = 6 \times {10^{ - 4}}Cm$
1. Now, calculating the value of electrical field from the equation of torque, by putting value of dipole moment in the torque equation:
$\tau = \vec p \times \vec E = pE\sin \theta$
Where,
$\tau =$ Torque
$\vec p =$Dipole Moment
$\theta = {30^o}$
Given, $\tau = 6\sqrt 3 Nm$
Substituting the known values in the equation:
$6\sqrt 3 Nm = 6 \times {10^{ - 4}}Cm \times \vec E$
$\vec E = 2\sqrt 3 N/C$
2. Now, putting the value of electrical field in the equation of potential energy:
$U = \vec p \cdot \vec E = - pE\cos \theta$
$\vec p =$Dipole Moment
$\vec E =$Electric Field
$U =$Potential Energy
$U = - (6 \times {10^{ - 4}}) \times (2\sqrt 3 \times {10^4}) \times \cos {60^o}$
$\Rightarrow U = - 18J$
Potential Energy is denoted by a negative sign, the charge is moved against the direction of the electric field.
This is our required solution.

Note: Electrical Dipole Moment is a quantity of measure of separation of two opposite charges separated by a distance. Dipole moment is a vector quantity.
Electric field can be defined as force exerted per unit charge. This is also a vector quantity.
Potential Energy can be defined as the amount of energy required to move a charge against the electric field.