Question

A point charge \(q_1 = +\)2.40 \(\mu\)C is held stationary at the origin. A second point charge \(q_2 = -\)4.30 \(\mu\)C moves from the point \(x =\) 0.150 m, \(y =\) 0 to the point \(x =\) 0.250 m, \(y =\) 0.250 m. How much work is done by the electric force on \(q_2\)?

Short answer

The work done by the electric force is approximately -3.56 Joules.

Step-by-step solution

Understand the Problem

We need to calculate the work done by the electric force as charge \( q_2 \) moves from one point to another in the electric field of charge \( q_1 \). The two charges have values of \( q_1 = +2.40 \mu C \) and \( q_2 = -4.30 \mu C \).

Initial and Final Positions

The initial position of \( q_2 \) is \((x_1, y_1) = (0.150, 0)\) and the final position is \((x_2, y_2) = (0.250, 0.250)\).

Calculate Initial Distance

Calculate the distance \( r_1 \) between \( q_1 \) and the initial position of \( q_2 \). Since \( q_1 \) is at the origin, \( r_1 = \sqrt{(0.150)^2 + 0^2} = 0.150 \) m.

Calculate Final Distance

Calculate the distance \( r_2 \) between \( q_1 \) and the final position of \( q_2 \). Since \( q_2 \) moves to point \((0.250, 0.250)\), \( r_2 = \sqrt{(0.250)^2 + (0.250)^2} = \sqrt{0.125} \simeq 0.354 \) m.

Use Formula for Work Done

The work done \( W \) is given by the formula: \[ W = k \cdot q_1 \cdot q_2 \left( \frac{1}{r_2} - \frac{1}{r_1} \right) \] where \( k = 8.99 \times 10^9 \; N \cdot m^2/C^2 \).

Substitute Values and Calculate

Substitute the known values into the formula: \[ W = 8.99 \times 10^9 \times 2.40 \times 10^{-6} \times (-4.30 \times 10^{-6}) \left( \frac{1}{0.354} - \frac{1}{0.150} \right) \] Calculate \( W \).

Final Calculation

After calculating, \( W \approx -3.56 \, \text{Joules} \), indicating that the electric force does this amount of work on \( q_2 \) while moving between the two points.

Key concepts

Point Charge

In the world of electromagnetism, a "point charge" refers to an idealized model of a charged object that is assumed to be infinitely small. This is helpful in simplifying calculations involving electric fields and forces.
This charge is treated as a single point in space, with no dimensions, allowing us to focus solely on the effects of the charge itself, rather than its size or shape. In practice, actual charges always occupy space, but this simplification helps physicists make useful predictions and calculations more efficiently.
For example, in the problem provided, both charges ( - one at the origin, - the other moving through space), can be considered point charges. This allows us to calculate the force and work done by the electric field generated by these charges using straightforward mathematical formulas.

Work Done

"Work done" by an electric force refers to the energy transferred by this force when a charge moves in an electric field. When a charge moves from one point to another, the electric force can do work on the charge, which can either be positive or negative depending on the movement direction relative to the force.
In the example exercise, we initially identify two positions for our second charge ( - an initial position at point (0.150, 0) m - and a final position at (0.250, 0.250) m). The force exerted by the electric field does work on the moving charge as it transitions between these locations.
This work done ( - calculated using the initial and final distances between charges), provides insight into how the energy of the system changes during the charge's movement. In our solution, the calculation showed that the work done by the electric force was approximately -3.56 Joules.

Electric Field

The "electric field" is a region around a charged object where other charges experience a force. This concept is crucial for understanding how charges interact in space.
An electric field is depicted as a vector field, which means it has both a magnitude and a direction at every point in space. The direction of the field at any point is the direction of the force that a positive test charge would experience at that point.
In our example, - the electric field created by point charge 1 influences the motion of point charge 2. As charge 2 moves, it experiences a force due to the electric field, which results in work being done, as previously discussed. Understanding how electric fields interact with charges is fundamental to solving problems involving electrostatic forces and energies.

Coulomb's Law

"Coulomb's Law" is a fundamental principle used to calculate the electric force between two point charges. It states that the force (- either attractive or repulsive)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 mathematical equation for Coulomb's Law is:\[F = k \frac{|q_1 q_2|}{r^2}\]where:

• \(F\) is the magnitude of the force between the charges,
• \(k\) is Coulomb's constant (\(8.99 \times 10^9\, N \cdot m^2/C^2\)),
• \(|q_1|\) and \(|q_2|\) are the magnitudes of the charges,
• \(r\) is the distance between the charges.

In the widget exercise, Coulomb's Law helps us calculate the initial and final forces, thus allowing us to determine the work done as one charge moves in the electric field created by another charge.