Animated Solution for Physics - Electrostatics: An electric dipole is placed at an angle of $30^{\circ }$ to a non-uniform electric field. The dipole will experience

Dipole in a Non-Uniform Field

Imagine an electric dipole placed in a non-uniform electric field. Unlike a uniform field where the electric field lines are perfectly parallel and equally spaced, a non-uniform field has lines that either converge or diverge.

Notice how the electric field lines in our setup are closer together on the left and spread out on the right. This spatial variation is crucial because it means the electric field strength $\mathbf{E}$ is stronger on the left side and weaker on the right side.

Unequal Forces

Now, let's look at the forces acting on the dipole. The dipole consists of a positive charge $+q$ and a negative charge $-q$. The positive charge experiences a force ${\mathbf{F}}_2=+q{\mathbf{E}}_2$ in the direction of the local electric field, while the negative charge experiences a force ${\mathbf{F}}_1=-q{\mathbf{E}}_1$ in the opposite direction.

Because the field is non-uniform, the value of the electric field at the location of the negative charge (${\mathbf{E}}_1$) is different from that at the positive charge (${\mathbf{E}}_2$). Therefore, the magnitudes of the two forces will also be unequal ($|\mathbf{F}_1| eq |\mathbf{F}_2|$).

This creates an imbalance. The net force ${\mathbf{F}}_{net}={\mathbf{F}}_1+{\mathbf{F}}_2$ will not be zero, meaning the dipole will experience a net translational force and will accelerate linearly.

Torque

Furthermore, the dipole is placed at an angle of $30^{\circ }$ to the general direction of the field. Because of this angle, the lines of action of the two forces ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$ do not coincide.

Whenever you have two forces acting on a rigid body along different lines of action, they create a turning effect. They form a couple, which means a net torque ${\mathbf{\tau }}_{net}$ will also act on the dipole, causing it to rotate.

So, what happens when you place a dipole in a non-uniform electric field at an angle? It doesn't just sit there, and it doesn't just spin. It does both! The unequal forces drag it across space (translational force), and the offset lines of action twist it around its center (torque). Therefore, the dipole experiences both a torque and a translational force.