Question

Three identical spheres each having a charge \(\mathrm{q}\) and radius \(R\), are kept in such a way that each touches the other two spheres. The magnitude of the electric force on any sphere due to other two is \(\ldots \ldots \ldots\) (A) \((\mathrm{R} / 2)\left[1 /\left(4 \pi \epsilon_{0}\right)\right](\sqrt{5} / 4)(\mathrm{q} / \mathrm{R})^{2}\) (B) \(\left[1 /\left(8 \pi \epsilon_{0}\right)\right](\sqrt{2} / 3)(\mathrm{q} / \mathrm{R})^{2}\) (C) \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right](\sqrt{3} / 4)(\mathrm{q} / \mathrm{R})^{2}\) (D) \(-\left[1 /\left(8 \pi \epsilon_{0}\right)\right](\sqrt{3} / 2)(\mathrm{q} / \mathrm{R})^{2}\)

Short answer

The magnitude of the electric force on any sphere due to the other two spheres is \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right](\sqrt{3} / 4)(q / R)^{2}\).

Step-by-step solution

Identifying Information

From the given problem, we know that each sphere has a charge q and radius R.

Determine the Distance Between the Charges on Each Sphere

Since the spheres touch each other, the distance between the centers of any two spheres is the sum of their radii. As their radii are equal (\(R\)), the distance between the charges on two spheres is: \(d = 2R\)

Calculate Electric Force Between One Pair of Spheres

Using Coulomb's Law, find the electric force (\(F_{12}\)) between two spheres. Coulomb's Law states that the force between two charges is given by: \(F=k \cdot \frac{q_1 \cdot q_2}{r^2}\), where \(k=\frac{1}{4 \pi \epsilon_0}\) Since both charges are equal (q), the force on any one sphere (1) due to the other sphere (2) is: \(F_{12}=\left(\frac{1}{4 \pi \epsilon_{0}}\right)\frac{q^2}{(2R)^2}\)

Calculate the Net Electric Force on One Sphere

Given three spheres forming a triangle, let us label the spheres A, B, and C. To find the net electric force on sphere A, we need to sum the electric forces acting on it, i.e., forces due to sphere B and sphere C. First, find the angle \(\theta\) between \(F_{AB}\) and \(F_{AC}\). Since they form an equilateral triangle, \(\theta= 60^{\circ}\) Now, calculate the net force by adding forces \(F_{AB}\) and \(F_{AC}\) in a vector form. Considering that both forces have a magnitude of \(F_{12}\), the net force can be determined using the cosine rule as: \(F_A = \sqrt{{F_{12}}^2 + {F_{12}}^2 + 2 \cdot F_{12} \cdot F_{12} \cdot \cos{60^{\circ}}}\)

Simplify the Net Electric Force Formula

Substitute \(F_{12}\) and \(\cos{60^{\circ}} = \frac{1}{2}\) in the above equation and simplify: \(F_A = \sqrt{{\left(\frac{1}{4 \pi \epsilon_{0}}\frac{q^2}{(2R)^2}\right)}^2 + {\left(\frac{1}{4 \pi \epsilon_{0}}\frac{q^2}{(2R)^2}\right)}^2 + 2 \cdot \left(\frac{1}{4 \pi \epsilon_{0}}\frac{q^2}{(2R)^2}\right) \cdot \left(\frac{1}{4 \pi \epsilon_{0}}\frac{q^2}{(2R)^2}\right) \cdot \frac{1}{2}}\) After simplifying, we obtain: \(F_A = \left[1 /\left(4 \pi \epsilon_{0}\right)\right](\sqrt{3} / 4)(q / R)^{2}\) Comparing with the given options, the magnitude of the electric force on any sphere due to the other two spheres is answer choice (C).

Key concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length, and all three interior angles are equal, each measuring 60 degrees. This symmetry makes calculations involving forces much more manageable. In geometry and physics, specifically in situations involving identical particles or charges, the equilateral triangle plays a significant role.

When three charged spheres are positioned such that each one touches the others, they naturally form the vertices of an equilateral triangle. This formation occurs because each side of the triangle equals the sum of two radii (in our case, two radii equal the side length).

Key properties of equilateral triangles:

• All sides are of equal length.
• All interior angles are 60 degrees.
• Symmetrical distribution of points or forces around the center.

Such properties simplify many calculations in physics and engineering, including those involving electric forces like in the problem at hand.

Electric Force Calculation

To calculate the electric force between two charged objects, we use Coulomb's Law. This law tells us that the electric force (F) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula for this law is:

\[ F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} \]

Where:

• \(F\) is the magnitude of the force between the charges.
• \(k\) is Coulomb's constant, \(\frac{1}{4 \pi \epsilon_0}\).
• |\(q_1\)| and |\(q_2\)| are the magnitudes of the charges.
• \(r\) is the distance between the charges.

Applying this to our scenario with spheres placed at corners of an equilateral triangle, the distance \(r\) between each pair of touching spheres is twice the radius \(R\) of a single sphere, resulting in each having a distance of \(2R\).

After calculating the force \( F_{12} \) between a pair, the symmetrical nature of the equilateral triangle simplifies the vector addition of forces.

Superposition Principle

The superposition principle is a foundational concept in physics that states that when multiple forces act on a body, the total force is the vector sum of all individual forces. This principle is essential for solving problems involving multiple charges and forces, such as the one we have with three charged spheres.

In our exercise, each sphere experiences forces due to the other two spheres, positioned at the vertices of an equilateral triangle. By applying the superposition principle, we can find the net force acting on one sphere by considering the forces due to the other two spheres individually and then adding them as vectors.

Here’s how superposition aids this problem:

• Calculate the force between one sphere and another using Coulomb's Law (e.g., \(F_{AB}\) between spheres A and B).
• Repeat for the force from the third sphere (e.g., \(F_{AC}\) between spheres A and C).
• Use vector addition to find the resultant force on the sphere (in this case, sphere A).

Because the spheres form an equilateral triangle, the angle between the forces is 60 degrees, which can be utilized in the vector addition formula. This scenario reveals the elegant simplicity that the superposition principle brings to multi-force physics problems.