Two small plastic spheres are given positive electric charges. When they are $15.0$ cm apart, the repulsive force between them has magnitude $0.220$ N. What is the charge on each sphere if the two charges are equal?

Verified step by step guidance

1. Start by identifying the formula for the electric force between two point charges, which is given by Coulomb's Law: $F=\frac{k{q}_1{q}_2}{r^2}$, where $F$ is the force, $k$ is Coulomb's constant ($8.99\times 109^{}$), $q_1$ and $q_2$ are the charges, and $r$ is the distance between the charges.
2. Since the problem states that the two charges are equal, we can set $q_1$ = $q_2$ = $q$. Substitute this into Coulomb's Law to get: $F=\frac{k{q}^2}{r^2}$.
3. Rearrange the formula to solve for $q^2$: $q^2=\frac{F{r}^2}{k}$. Then, take the square root of both sides to find $q$: $q=\sqrt{\frac{F{r}^2}{k}}$.
4. Substitute the given values into the equation: $F$ = 0.220 N, $r$ = 0.15 m, and $k$ = $8.99\times 109^{}$. Plug these values into the formula to find $q$.
5. Calculate the expression to find the charge $q$ on each sphere. Remember to keep track of units and ensure they are consistent throughout the calculation.

Key Concepts

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electrostatic force between two 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 is F = k * |q1 * q2| / r^2, where k is Coulomb's constant.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. Charges can be positive or negative, and like charges repel each other while opposite charges attract. In this problem, both spheres have positive charges, leading to a repulsive force between them.

Equilibrium of Forces

In physics, equilibrium refers to a state where all forces acting on an object are balanced, resulting in no net force and no acceleration. For the spheres, the repulsive force due to their charges must be equal and opposite to any other forces acting on them, ensuring they remain stationary or move uniformly.