Two small spheres spaced $20.0$ cm apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is $3.33\times {10}^{-21}$ N?

Verified step by step guidance

1. Start by understanding that the force between two charged objects is given by Coulomb's Law, which is expressed as: $F=\frac{k{q}^2}{r^2}$, where $F$ is the force, $k$ is Coulomb's constant ($8.99\mathrm{\times }109^{}$), $q$ is the charge on each sphere, and $r$ is the distance between the spheres.
2. Rearrange Coulomb's Law to solve for the charge $q$: $q=\sqrt{\frac{F{r}^2}{k}}$. Substitute the given values: $F$ = 3.33 × 10^-21 N and $r$ = 0.20 m.
3. Calculate the charge $q$ using the rearranged formula. This will give you the charge in coulombs on each sphere.
4. Understand that the charge of an electron is approximately $-1.60\mathrm{\times }10{-19}^{}$. To find the number of excess electrons, divide the calculated charge $q$ by the charge of a single electron.
5. Perform the division to find the number of excess electrons on each sphere. This will give you the number of electrons needed to produce the calculated charge.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The formula is F = k * (|q1 * q2| / r^2), where F is the force, q1 and q2 are the charges, r is the distance, and k is Coulomb's constant.

Charge of an Electron

The charge of an electron is a fundamental constant in physics, approximately -1.602 x 10^-19 coulombs. This value is crucial for calculating the number of excess electrons on an object, as it allows us to determine the total charge by multiplying the number of electrons by the charge of a single electron.

Electric Charge Quantization

Electric charge quantization refers to the principle that charge comes in discrete amounts, specifically integer multiples of the elementary charge (e.g., the charge of an electron or proton). This concept is essential for understanding that the total charge on an object is the sum of individual charges, which are quantized in units of the electron's charge.