Animated Solution for Physics - Electric Charges and Fields: Two identical tennis balls each having mass $m$ and charge $q$ are suspended from a fixed point by threads of length $l$. What is the equilibrium separation when each thread makes a small angle $\theta$ with the vertical ?

The problem of two charged pendulums is a classic in physics, beautifully marrying the principles of Newtonian mechanics with Coulomb's law of electrostatics. Let's dive deep into the forces at play and unravel the math step-by-step.

Imagine two identical tennis balls, each carrying a mass $m$ and a charge $q$. They are suspended from a single rigid point by two massless threads, each of length $l$. Because both balls carry the exact same charge $q$, they experience a mutual electrostatic repulsion. This invisible force pushes them apart until they reach a state of perfect equilibrium, separated by a distance $d$. At this equilibrium, each thread makes a small angle $\theta$ with the vertical axis.

Our goal is to find an expression for this equilibrium separation $d$ in terms of the given variables.

To understand the equilibrium, we must isolate one of the balls—let's pick the left one—and draw its free body diagram. There are exactly three forces acting on this ball:

1. Gravity: The Earth pulls the ball downwards with a force equal to its weight, $mg$. 2. Electrostatic Repulsion: The right ball pushes the left ball horizontally outward with a Coulomb force, $F$. 3. Tension: The thread pulls the ball upward and inward along its length with a tension force, $T$.

Because the tension $T$ acts at an angle $\theta$ to the vertical, it's most helpful to resolve it into vertical and horizontal components. The vertical component, adjacent to the angle, is $T\cos \theta$. The horizontal component, opposite to the angle, is $T\sin \theta$.

Since the ball is resting in perfect equilibrium, the net force acting on it must be zero in all directions. This gives us two fundamental equations:

Balancing the vertical forces:

Balancing the horizontal forces:

We are looking for the separation $d$, and the tension $T$ is just an internal reaction force that we don't explicitly need. The most elegant way to eliminate $T$ is to divide the horizontal equation by the vertical equation:

This simplifies beautifully to:

Now, we bring in the electrostatics. According to Coulomb's law, the repulsive force $F$ between two charges $q$ separated by a distance $d$ in a vacuum is given by:

Substituting this expression for $F$ back into our equilibrium equation yields:

Here is where the physics intuition kicks in. The problem explicitly states that the angle $\theta$ is "small". In mathematics, when an angle $\theta$ is very small, the tangent of the angle is approximately equal to the sine of the angle ($\tan \theta \approx \sin \theta$).

If we look at the right-angled triangle formed by the thread, the vertical axis, and half the separation distance, we can write the exact expression for $\sin \theta$:

Therefore, using our small angle approximation, we can state:

We now have two different expressions for $\tan \theta$. By equating them, we can solve for our target variable, $d$:

To isolate $d$, we cross-multiply. Multiplying both sides by $d^2$ gives $d^3$ on the left side. Moving $2l$ to the numerator on the right side gives:

The $2$ in the numerator partially cancels the $4$ in the denominator, leaving a $2$:

Finally, to find $d$, we take the cube root of both sides:

This perfectly matches option (b). The beauty of this problem lies in how a complex interplay of forces simplifies into an elegant algebraic expression through the strategic use of approximations.