Problem 30 - College Physics

Question

(a) \(A+5.00\) pC charge is located on a sheet of paper. (a) Draw to scale the curves where the equipotential surfaces due to these charges intersect the paper. Show only the surfaces that have a potential (relative to infinity) of \(1.00 \mathrm{V}, 2.00 \mathrm{V}\) \(3.00 \mathrm{V}, 4.00 \mathrm{V},\) and 5.00 \(\mathrm{V} .\) (b) The surfaces are separated equally in potential. Are they also separated equally in distance? (c) In words, describe the shape and orientation of the surfaces you just found.

Step-by-Step Solution

Answer

Equipotential surfaces are concentric circles, spaced closer with increasing potential, due to decreasing radius with increasing potential.

Step 1: Understand the Problem

We have a point charge of +5.00 picocoulombs (pC) on a sheet. We need to visualize equipotential surfaces (curves, since we're viewing on a 2D plane) for potentials of 1, 2, 3, 4, and 5 volts. The question also asks whether these surfaces are evenly spaced in distance and the description of their shape and orientation.

Step 2: Recall the Formula for Electric Potential

The electric potential \( V \) due to a point charge is given by: \[ V = \frac{k \, Q}{r} \]where \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, ext{Nm}^2/ ext{C}^2 \)), \( Q \) is the charge, and \( r \) is the distance from the charge. We will invert this formula to find \( r \) for each potential.

Step 3: Calculate Distance for Each Potential

For each potential \( V = 1.00, 2.00, 3.00, 4.00, 5.00 \, ext{V} \), find the distance \( r \). Using \( Q = 5.00 \, ext{pC} = 5.00 \times 10^{-12} \, ext{C} \):1. For \( V = 1.00 \, ext{V} \):\[ r = \frac{k \, Q}{V} = \frac{8.99 \times 10^9 \, ext{Nm}^2/ ext{C}^2 \times 5.00 \times 10^{-12} \, ext{C}}{1} \, ext{V} = 0.04495 \, ext{m} \]2. Repeat for \( V = 2.00, 3.00, 4.00, 5.00 \, ext{V} \) to get respective \( r \) values.

Step 4: Draw the Equipotential Curves

Using the distances calculated for each potential, draw circles (equipotential lines) with radii corresponding to the calculated distances: 0.04495 m for 1 V, subsequently decreasing for higher potentials. Note that these circles will appear closer together as potential increases.

Step 5: Analyze Distance Between Equipotential Surfaces

The potential difference is equal for each surface, but the distances between these surfaces are not equal. Equipotential surfaces are closer together as the potential increases because \( r \) decreases with \( V \).

Step 6: Describe Shape and Orientation

The surfaces are spherical in 3D but appear as concentric circles when viewed on the paper. The charge is at the center, and the surfaces are oriented outward from the charge, showing decreasing radius for increasing potential values.

Key Concepts

Equipotential Surfaces

Equipotential surfaces are imaginary planes where the electric potential is the same throughout. They serve as a visualization tool to understand electric field interactions. In the context of a point charge, these surfaces appear as concentric spheres in three-dimensional space, with the point charge at their center. However, when viewed on a two-dimensional plane, such as a sheet of paper, these surfaces become concentric circles.

• Each circle represents a point where the electric potential is identical.
• As the distance from the charge increases, the potential decreases, leading to surfaces that grow in size.
• The thickness of these surfaces decreases as we move outward from the charge.

Equipotential surfaces are helpful because they illustrate that no work is required to move a charge along the surface. This is because electric field lines intersect equipotential surfaces at right angles.

Point Charge

A point charge is a charge that is so small it can be considered a single point in space. Mathematically, it is treated as having no dimensions, focused at a single location. Point charges are an essential concept in electromagnetism as they allow us to simplify complex charge distributions into manageable models.

• Point charges are often used in theoretical problems to derive formulas for electric fields and potentials.
• The behavior of point charges is governed by Coulomb's law, which describes the forces between charges.
• In this exercise, a +5.00 pC charge acts as a point charge creating electric potentials around itself.

The concept of a point charge allows us to predict how electric fields behave in the vicinity of charged objects.

Coulomb's Constant

Coulomb's constant, denoted as \( k \), is a fundamental constant in physics that appears in Coulomb's law. It characterizes the strength of the electric force between two point charges. Prevalent in the equations of electromagnetism, it is valued at \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \). This constant facilitates the calculation of electric potential and fields.

• \( k \) is essential for determining force interactions between electrically charged particles.
• It is derived from the permittivity of free space, which is a measure of how much electric field permeates a vacuum.
• In practical calculations, Coulomb's constant simplifies the relationship between force, distance, and charge.

Remember that while \( k \) is always constant, the forces and potentials it helps calculate can change based on the specific charges and distances involved.

Distance vs Potential Relationship

The relationship between distance and electric potential due to a point charge is inversely proportional. This means that as distance \( r \) from the charge increases, the potential \( V \) decreases. The formula describing this relationship is \( V = \frac{kQ}{r} \). Here, \( k \) is Coulomb's constant, and \( Q \) is the point charge.

• Potential is greater closer to the charge and decreases with distance.
• As seen in the exercise, surfaces of equal potential (equipotentials) are closer together near the charge.
• The change in potential per unit distance decreases as we move further from the charge.

This inverse relationship outlines why equipotential surfaces become denser around a charge, offering a clear visual representation of how electric fields diminish with distance.