Problem 25 - College Physics

Question

(a) The equipotential surfaces in the neighborhood of a positive point charge are spheres. Which sphere is associated with the higher electric potential: (1) the smaller one, (2) the larger one, or (3) they are associated with the same potential? (b) Calculate the amount of work (in electron-volts) it would take to move an electron from \(12.6 \mathrm{~m}\) to \(14.3 \mathrm{~m}\) away from \(\mathrm{a}+3.50-\mu \mathrm{C}\) point charge.

Step-by-Step Solution

Answer

(a) The smaller sphere is at higher potential. (b) The calculated work is approximately 45.5 eV.

Step 1: Understanding Equipotential Surfaces

Equipotential surfaces are surfaces around a charge where the electric potential is constant. For a positive point charge, these surfaces are concentric spheres centered on the charge. The potential is higher closer to the charge. Thus, the smaller sphere, being closer, is at a higher potential compared to a larger sphere further away.

Step 2: Determining Which Sphere is at Higher Potential

Since the electric potential \(V\) due to a point charge is given by \(V = \frac{kQ}{r}\), where \(k\) is a constant, \(Q\) is the charge, and \(r\) is the distance from the charge, smaller \(r\) implies higher potential. Therefore, the smaller sphere is associated with a higher potential than the larger one.

Step 3: Calculation of Work Required

The work done (W) in moving a charge in an electric field is the difference in potential energy: \(W = q(V_f - V_i)\), where \(q\) is the charge moved, \(V_i\) and \(V_f\) are the initial and final potentials. For the work to move an electron (charge \(-e\)) from 12.6 m to 14.3 m from a \(+3.50 \, \mu C\) point charge, calculate potentials at both distances.

Step 4: Calculate Initial and Final Potentials

Using \(V = \frac{kQ}{r}\), compute \(V_i = \frac{(8.99 \times 10^9 \, N\cdot m^2/C^2)(3.50 \times 10^{-6} \, C)}{12.6 \, m}\) and \(V_f = \frac{(8.99 \times 10^9 \, N\cdot m^2/C^2)(3.50 \times 10^{-6} \, C)}{14.3 \, m}\).

Step 5: Compute the Work Done in Electron-Volts

Calculate the work: \(W = -e(V_f - V_i)\). Convert the result from joules to electron-volts using 1 eV = \(1.602 \times 10^{-19} \, J\). For an electron, \(e \, is \, -1.60 \times 10^{-19} \, C\).

Step 6: Substitute and Solve for Work

Substitute the obtained potentials: \(W = (-1.60 \times 10^{-19} \, C)((8.99 \times 10^9 \times 3.50 \times 10^{-6} / 14.3) - (8.99 \times 10^9 \times 3.50 \times 10^{-6} / 12.6))\), results in work done. Convert it to eV.

Key Concepts

Equipotential Surfaces

Equipotential surfaces are fascinating features in the world of electrostatics. Think of them as invisible layers or shells that envelop a charge. These surfaces have a unique trait: the electric potential is the same at every point on the surface.
For a positive point charge, these surfaces appear as concentric spheres surrounding the charge. Picture a tiny positive charge sitting at the center and its influence extending outward like layers of an onion. The closer you are to the charge, the higher the potential.

• The smaller the sphere, the higher the potential.
• The larger the sphere, the lower the potential.

Electric Potential

Electric potential is a measure of the potential energy per unit charge exerted by an electric field on a point charge. It tells us how much work would be required to move a charge to a point within the field.

For a single point charge, the electric potential (V) is described by the equation: \( V = \frac{kQ}{r} \).

• \(k\) is Coulomb's constant, roughly \(8.99 \times 10^9 \, \text{N}\cdot \text{m}^2/\text{C}^2\).
• \(Q\) is the magnitude of the charge creating the field.
• \(r\) is the radial distance from the charge to the point of interest.

A higher electric potential indicates more energy per unit charge is present at that point. It's essential to remember that electric potential is a scalar quantity, meaning it does not have a direction.

Point Charge

A point charge is a theoretical concept commonly used in physics to simplify problems involving electric charges. It's considered to have all its charge concentrated at a single point in space, which allows us to easily consider its effects without worrying about its actual size.

In this exercise, the positive charge defined is quite small, \(+3.50\, \mu\text{C}\), yet it creates equipotential surfaces that are spheres. By understanding the impact of a point charge, we can predict how the electric field and potential behave as we move away from this charge.

• A stronger charge results in a stronger electric field and higher potentials close to it.
• As you move further away, the influence decreases following an inverse relationship with the distance.

Work Calculation

Work in the context of electrostatics refers to the energy required to move a charge within an electric field. The work calculation involves finding the change in potential energy between two points and multiplying it by the charge moved.

The equation used is \( W = q(V_f - V_i) \).Here the challenge is to find the work done on an electron, which is negatively charged, when moving between two distances (\(12.6 \, m\)and\(14.3 \, m\))from a \(+3.50\, \mu\text{C}\)point charge.

To determine the potentials (\(V_i\)and \(V_f\)), you use the formula:
\[ V = \frac{kQ}{r} \]where you substitute the initial and final distances into \(r\). After calculating the initial and final potential, you substitute them into the work equation. The charge (\(q\))for an electron is \(-1.60 \times 10^{-19} \, C\).To transform your result to electron-volts, use the conversion factor \(1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}\).