Q.Show that the torque acting on an electric dipole placed in a uniform electric field $\vec{E}$ is $\vec{\tau} = \vec{p}\times\vec{E}$, where $\vec{p}$ and $\vec{\tau}$ represent the dipole moment and torque respectively. OR What is resistance of a resistor? Show that the equivalent resistance of three resistors connected in series is equal to the sum of the resistances of the individual resistors.

Torque on a Dipole — From Intuition to the Formula

Imagine a bar magnet placed in a uniform magnetic field. You know that the north pole gets pulled one way and the south pole the opposite way. If the magnet is not aligned with the field, these two equal and opposite forces create a twist — a torque — that tries to rotate the magnet until it lines up with the field. That's the core idea.

The same thing happens with an electric dipole (two equal and opposite charges $+q$ and $-q$ separated by a small distance $d$) placed in a uniform electric field $\vec{E}$. The two charges experience forces in opposite directions, and unless the dipole is already parallel to the field, those forces produce a torque.

Step 1: The Forces on the Two Charges

Let the dipole moment $\vec{p}$ point from the negative charge to the positive charge, with magnitude $p = qd$.

In a uniform electric field $\vec{E}$:

• The positive charge $+q$ feels a force $\vec{F}_+ = +q\vec{E}$ (in the direction of $\vec{E}$).
• The negative charge $-q$ feels a force $\vec{F}_- = -q\vec{E}$ (opposite to $\vec{E}$).

These two forces are equal in magnitude but opposite in direction. They form a couple — a pair of equal, opposite, parallel forces that do not share the same line of action. A couple always produces a pure torque, with no net force.

Step 2: Why a Torque Appears

If the dipole is at an angle $\theta$ to the field, the two forces are not along the same line. They are separated by the perpendicular distance between their lines of action. That perpendicular distance is $d \sin\theta$, where $d$ is the separation between the charges.

The torque $\tau$ due to a couple is:

Here:

• Force magnitude on each charge: $F = qE$
• Perpendicular distance: $d \sin\theta$

So:

But $qd = p$, the magnitude of the dipole moment. Therefore:

Step 3: The Vector Form

Torque is a vector — it has a direction. The direction of the torque is perpendicular to both $\vec{p}$ and $\vec{E}$, following the right-hand rule. The complete vector equation is:

The magnitude is $|\vec{\tau}| = pE \sin\theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$.

Step 4: What the Torque Does

• When $\theta = 0^\circ$ (dipole aligned with the field): $\sin 0 = 0$, so $\tau = 0$. The dipole is in stable equilibrium — if you nudge it slightly, the torque brings it back.
• When $\theta = 90^\circ$ (dipole perpendicular to the field): $\sin 90^\circ = 1$, so torque is maximum: $\tau_{\text{max}} = pE$. …