Torque on a dipole in a uniform electric field

A charge generates an electric field around it. When another charge is near this charge, it creates electric field lines. If the charges are equal but opposite, then they form a dipole. There is a distance between the charges in the dipole. Hence, the dipole also has a dipole moment. 

There is a torque on a dipole in an uniform electric field. So, if you want to know more on this topic, read this entire article.

Definition of uniform electric field, dipole, and torque

Before beginning torque on a dipole in a uniform electric field, here are definitions of the basic terms of this topic.

Uniform electric field

All point charges create a field around it called an electric field. The charge creates a field when kept between two charged plates that are equidistant. At this point, the charge will create electric field lines parallel to each other. This situation is called a uniform electric field.

Dipole

A dipole has two point-sized opposite charges equal in magnitude and separated by a distance. 

Torque

Torque is a force that causes any two objects to rotate. The forces are equal in magnitude and opposite in direction in a dipole in a uniform field. These forces act at different points since a distance in a dipole separates the charges.

Dipole moment

Electric dipole moment refers to the product of charge q and the distance between the charges in a dipole. It is a vector quantity. The direction of the dipole moment goes along the line of the charges. It is given by,

P = q x 2a P̂

where;

P = electric dipole moment

q = charge

2a = distance between charges

P̂ = direction of the dipole moment

Torque on a Dipole in a Uniform Electric Field

Suppose a permanent dipole is in a uniform electric field. The permanent dipole has a dipole moment. The dipole is permanent because the dipole moment is present even without an electric field. Let the two charges in the dipole be q and -q. Let E be the electric field. The dipole moment is p.

Consider a force qE on charge q and -qE on charge -q. Since the dipole is in an electric field, the net force on the dipole is zero. But the charges are separated by a distance. So, the force acts at different points. It creates torque on the dipole. Since the forces act in opposite directions, the torque is a couple. 

When the net force is zero, this torque acting on a dipole is independent of its origin. The magnitude of the torque is equal to the product of the magnitude of each force and the arm of the couple. The arm is the perpendicular distance between the two antiparallel forces. The equation is as follows:

τ = q E x 2 a sin θ 

τ = q E 2 a sin θ 

Where;

τ = magnitude of the torque

q = charge

E = electric field

2 a = distance between charges

The direction of the torque on a dipole in a uniform electric field is perpendicular to the plane of the paper. It could be imagined as a line coming out of the paper at  90˚ angle. Since the value of sin 90˚ is 1, the magnitude of torque will become,

τ = q E 2 a 

But, p = q x 2 a

So, τ = p x E

Since the torque will rotate the charges, at some point, τ will align with E. Then, the angle will be 180˚. The value of sin 180˚ is 0. Hence, the torque τ, will be zero. 

Electric dipole in a non-uniform electric field

When the electric field is non-uniform, the net force on the dipole will not be zero. There will be a torque on the system already present. When the dipole moment is parallel to electric field E or it is antiparallel to E, the net torque acting on a dipole will be zero. But there will be a net force on the dipole if the electric field is non-uniform.

Everyday observation of torque on a dipole in electric field

A glass rod rubbed with silk attracts a plastic straw. The glass rod gets charged when run rubbed with silk by friction. The uncharged plastic straw moves towards the glass rod. The glass rod produces a net dipole moment in the direction of the electric field created by charges on the glass rod. This electric field is non-uniform. 

So, the dipole experiences a net force on it. For example, the glass rod is said to polarise the plastic straw. The force attracts the plastic straw in the direction of the glass rod. A few more similar examples are as follows.

• Paper pieces attracted by a comb rubbed on dry hair. As a result, the comb experiences a net force on it. So, the paper pieces move in the direction of the rod.
• An iron piece of paper brought near a T.V or computer screen gets stuck to the screen.

Conclusion

A charge develops an electric field around it. When another equal and opposite charge separated by a distance is brought near, it will become a dipole. There will be a dipole moment between them. When this system is in a uniform electric field, it will cause rotation. The force responsible for this rotation is called the torque. 

The torque acting on a dipole is one of the many important topics in Physics, especially motion.