Problem 24 - College Physics

Question

If the radius of the equipotential surface of a point charge is \(10.5 \mathrm{~m}\) and is at a potential of \(+2.20 \mathrm{kV}\) (compared to zero at infinity), what are the magnitude and sign of the point charge?

Step-by-Step Solution

Answer

The charge is approximately \(2.57 \, \mu\text{C}\) and is positive.

Step 1: Understand the Formula

To find the charge of the point charge, we need to use the formula for electric potential due to a point charge, which is given by \( V = \frac{kQ}{r} \), where \( V \) is the potential, \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2) \), \( Q \) is the charge, and \( r \) is the distance from the charge. We need to rearrange this formula to solve for \( Q \).

Step 2: Rearranging the Formula

Rearranging the formula \( V = \frac{kQ}{r} \) to solve for the charge \( Q \), we get: \[ Q = \frac{Vr}{k} \]

Step 3: Substitute the Known Values

We know \( V = +2.20 \, \text{kV} = 2.20 \times 10^3 \, \text{V} \) and \( r = 10.5 \, \text{m} \). Substituting these values into the formula: \[ Q = \frac{(2.20 \times 10^3) \times 10.5}{8.99 \times 10^9} \]

Step 4: Calculate the Charge \( Q \)

Carrying out the calculations, we get: \[ Q = \frac{(2.20 \times 10^3) \times 10.5}{8.99 \times 10^9} = \frac{23100}{8.99 \times 10^9} \approx 2.57 \times 10^{-6} \, \text{C} \] The sign of the charge is positive since the potential is positive.

Key Concepts

Electric Charge

Electric charge is a fundamental property of certain particles that causes them to experience a force when in the presence of other charges. Charges can be positive or negative, and like charges repel each other, while opposite charges attract each other. This force interaction is a cornerstone of electromagnetism.

Charges are measured in coulombs (C). It's a fundamental unit in physics, representing the quantity of electricity transported in one second by a current of one ampere. Studying charges helps us understand phenomena like static electricity, the operation of batteries, and the behavior of particles in electric fields.

An object gets charged either by gaining or losing electrons, which have a negative charge. When a material loses electrons, it becomes positively charged. Conversely, when it gains electrons, it becomes negatively charged.

Coulomb's Law

Coulomb's Law is a central principle of electromagnetism that describes how the electric force between two charges depends on their respective magnitudes and the distance between them. The formula is expressed as:

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

where:

• \(F\) is the force between the charges,
• \(q_1\) and \(q_2\) are the magnitudes of the charges,
• \(r\) is the distance between the charges,
• \(k\) is Coulomb's constant, approximately equal to \(8.99 \times 10^9 \, \mathrm{Nm}^2/\mathrm{C}^2\).

This law establishes that the force increases with the amount of charge and decreases as the distance between charges increases. It's very similar in structure to Newton's law of gravitation, though it deals with electric forces rather than gravitational.Coulomb's law allows us to calculate the electric force and appreciate how charges interact, influencing phenomena at both atomic and macroscopic scales.

Equipotential Surfaces

Equipotential surfaces are imaginary surfaces around a charge, on which every point has the same electric potential. Think of it like contour lines on a map that indicate equal elevation. These surfaces play a crucial role in understanding electrostatics.

They can be visualized as spherical shells surrounding a point charge. On any such shell, the potential is constant, meaning that no work is needed to move a charge along the surface. If you picture concentric spheres around a point charge, the surface potential decreases as you move farther from the charge.

The idea of equipotential surfaces helps us simplify complex electric fields by essentially "slicing" the space around a charge with these constant-potential layers. This makes it easier to visualize and compute electric fields and potentials in systems involving multiple charges.

Moreover, because these surfaces are perpendicular to electric field lines, they provide insight into how electric fields are oriented in space and assist in solving potential-related problems.