Animated Solution for Physics - Electrostatics: A pendulum bob of mass 80 mg and carrying a charge of $2\times 10^{-8}\text{ C}$ is at rest in a horizontal uniform electric field of 20,000 V/m. Find the tension in the thread of the pendulum and the angle it makes with the vertical. (Take $g=9.8{\text{ ms}}^{-2}$)

Imagine a pendulum bob suspended from a rigid support. In a normal scenario, gravity pulls the bob straight down, and it hangs perfectly vertical. However, in this problem, we introduce a horizontal uniform electric field.

Because the bob carries a positive charge, this electric field exerts a horizontal force on it. This force pushes the bob to the side, causing it to deflect from the vertical.

Eventually, the bob settles into a new equilibrium position at an angle $\theta$. Our goal is to completely define this new state by finding the tension in the string and the angle of deflection.

To solve any mechanics problem involving equilibrium, our best tool is the free body diagram. Let's identify the forces acting on the bob in its deflected position.

First, we have the gravitational force, or weight, acting straight down. This is given by $mg$.

Second, we have the electric force acting horizontally to the right. This is given by $F_e=qE$.

Finally, we have the tension $T$ in the string, pulling the bob along the length of the string at an angle $\theta$ to the vertical.

Since the tension is acting at an angle, it's incredibly useful to resolve it into horizontal and vertical components. The vertical component is $T\cos \theta$, and the horizontal component is $T\sin \theta$.

Because the bob is in perfect equilibrium, the net force in both the horizontal and vertical directions must be exactly zero.

Balancing the vertical forces, the upward component of tension must equal the downward weight:

Balancing the horizontal forces, the leftward component of tension must equal the rightward electric force:

These two equations are the master keys to unlocking the entire problem.

We have a system of two equations with two unknowns: $T$ and $\theta$. Let's find the tension first.

A brilliant mathematical trick to eliminate the angle $\theta$ is to square both equations and add them together.

Squaring the vertical equation gives:

Squaring the horizontal equation gives:

Adding them together, we get:

Thanks to the Pythagorean identity, ${\sin }^2\theta +{\cos }^2\theta =1$, the angle vanishes completely! We are left with a beautiful, elegant expression for the tension:

Notice how the formula $T=\sqrt{(mg{)}^2+(qE{)}^2}$ is exactly the Pythagorean theorem. This is no coincidence! Because the gravitational force acts purely vertically and the electric force acts purely horizontally, they form the two legs of a right-angled triangle. The tension in the string must perfectly balance the resultant of these two forces. Therefore, the magnitude of the tension is exactly equal to the hypotenuse of this force triangle. This geometric interpretation gives us a powerful visual way to understand the math.

Now, let's substitute our known values. We must be extremely careful with units. The mass is given as $80\text{ mg}$, which must be converted to kilograms: $80\times 10^{-6}\text{ kg}$.

The weight is:

The electric force is:

Plugging these into our tension formula:

Calculating the square root, we find the final tension:

With the tension found, we can now determine the angle of deflection, $\theta$.

Let's return to our two master equations. Instead of squaring and adding, what if we divide them?

Dividing the horizontal equation by the vertical equation gives:

The tension $T$ cancels out beautifully, leaving us with the tangent of the angle:

This equation makes perfect intuitive sense. The angle depends entirely on the ratio of the horizontal push (electric force) to the vertical pull (gravity).

Similarly, the equation $\tan \theta =\frac{qE}{mg}$ has a direct geometric meaning. In our force triangle, the angle $\theta$ is the angle between the vertical weight vector and the hypotenuse (the resultant force). The tangent of this angle is the opposite side (the horizontal electric force) divided by the adjacent side (the vertical weight). It is deeply satisfying when the algebraic manipulation perfectly mirrors the physical geometry of the system!

Let's substitute the values we calculated earlier:

The powers of ten cancel out, simplifying the calculation:

To find the angle itself, we take the inverse tangent:

We have successfully completely defined the state of the pendulum. The tension in the string is $8.79\times 10^{-4}\text{ N}$, and it hangs at an angle of $27^{\circ }$ from the vertical.

This problem is a fantastic illustration of how different fundamental forces can interact. The gravitational field of the Earth and the applied uniform electric field work together to create a new, stable equilibrium.

Imagine if we were to slowly increase the electric field. The horizontal force would grow, pushing the bob further out, increasing the angle $\theta$, and simultaneously increasing the tension in the string. The physics is dynamic, interconnected, and deeply elegant!