Problem 19 - University Physics with Modern Physics

Question

Two point charges \(q_{1}=+2.40 \mathrm{nC} \quad\) and \(\quad q_{2}=\) \(-6.50 \mathrm{nC}\) are 0.100 \(\mathrm{m}\) apart. Point \(A\) is midway between them; point \(B\) is 0.080 \(\mathrm{m}\) from \(q_{1}\) and 0.060 \(\mathrm{m}\) from \(q_{2}\) (Fig. E23.19). Take the electric potential to be zero at infinity. Find (a) the potential at point \(A\); (b) the potential at point \(B ;\) (c) the work done by the electric field on a charge of 2.50 \(\mathrm{nC}\) that travels from point \(B\) to point \(A\) .

Step-by-Step Solution

Answer

(a) Use potential formula for both charges at midpoint; (b) Sum potentials at respective distances for each charge; (c) Calculate work using potential difference and charge.

Step 1: Calculate Potential at Point A

The electric potential at a point due to a point charge is given by the formula \( V = \frac{kq}{r} \), where \( k = 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge to the point. Since Point A is midway between the charges, the distance from each charge to Point A is \( r = 0.050 \, \mathrm{m}\) (half of 0.100 m). Calculate the potential due to each charge and then sum them: \( V_A = \frac{k q_1}{r} + \frac{k q_2}{r} \).

Step 2: Calculate Potential at Point B

For Point B, calculate the potential separately due to each charge at their respective distances and sum them. The potential at a point due to a charge is \( V = \frac{kq}{r} \). Hence: \( V_B = \frac{k q_1}{0.080} + \frac{k q_2}{0.060} \). Use the given distances 0.080 m for \( q_1 \) and 0.060 m for \( q_2 \).

Step 3: Calculate Work Done from B to A

The work done by the electric field on a charge \( q \) moving from a potential \( V_B \) to \( V_A \) is given by the formula \( W = q(V_A - V_B) \). Use the charge value \( q = 2.50 \, \mathrm{nC} = 2.50 \times 10^{-9} \, \mathrm{C} \) and the potentials calculated in Steps 1 and 2 to find the work done: \( W = 2.50 \times 10^{-9} (V_A - V_B) \).

Key Concepts

Coulomb's Law

Coulomb's law is a fundamental principle in physics that describes the force between two point charges. It states that the force (\( F \) ) between two charges is directly proportional to the product of the magnitudes of the charges (\( q_1 \) and \( q_2 \) ) and inversely proportional to the square of the distance (\( r \) ) between them. The mathematical expression is:\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]Where \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \) ).

• The force is attractive if the charges are of opposite signs and repulsive if the same.
• It is a vector quantity, meaning it has both magnitude and direction.

Understanding Coulomb's law helps to calculate electric potential, as potential is derived from the energy interactions described by this force law.

Point Charge

A point charge is an idealized electric charge that is considered to have no size; it is focused entirely at a single point in space. Point charges are used to simplify the analysis of electrical interactions in space.

• They are hypothetical concepts that represent charged particles, such as electrons or protons, with negligible size.
• Their influence on the electric field and potential is calculated using Coulomb's law.

In exercises involving point charges, the calculation of electric fields and potential around them assumes that these charges are not spread out but concentrated at distinct points, making the math easier to handle. This is crucial for understanding how charges interact at various distances.

Electric Field

The electric field is a vector field around a charged particle that represents the force exerted per unit charge at any point in the field. It is a fundamental concept when dealing with charges.The electric field (\( E \) ) created by a point charge (\( q \) ) at a distance (\( r \) ) is given by:\[ E = \frac{kq}{r^2} \]Where \( k \) is the Coulomb's constant.

• The direction of the electric field due to a positive charge is radially outward, and inward for a negative charge.
• Its strength decreases with the square of the distance from the charge.

The concept of an electric field allows us to understand how charges can influence other charges even at a distance, without any physical connection.

Work Done by Electric Field

The work done by the electric field is an important concept related to how energy is transferred to or from a charged particle as it moves within an electric field.The work done (\( W \) ) by an electric field on a charge (\( q \) ) as it moves from one point to another is calculated using the change in electric potential (\( \Delta V \) ) between those points:\[ W = q \cdot (V_A - V_B) \]Where \( V_A \) and \( V_B \) are the electric potentials at the initial and final points, respectively.

• If the charge moves along the direction of the field, the field does positive work and transfers energy to the charge.
• If the charge moves against the direction of the field, the field does negative work and energy is taken from the charge.

Understanding this concept helps in determining energy conversions in electrostatics, crucial for both theoretical analyses and practical applications.