Problem 20 - College Physics

Question

A uniform electric field of \(10 \mathrm{kV} / \mathrm{m}\) points vertically upward. How far apart are the equipotential planes that differ by \(100 \mathrm{~V} ?\)

Step-by-Step Solution

Answer

The distance is 1 cm.

Step 1: Understanding the Problem

We have a uniform electric field with a magnitude of \(10 \text{ kV/m}\) pointing upward. We need to find the distance between equipotential planes that have a potential difference of \(100\text{ V}\).

Step 2: Key Formula for Distance in Electric Field

The relationship between electric potential difference \(\Delta V\), electric field \(E\), and distance \(d\) is given by the formula: \(\Delta V = E \times d\). We can rearrange this formula to solve for \(d\): \(d = \frac{\Delta V}{E}\).

Step 3: Substituting Values

Substitute the given values into the formula: \(\Delta V = 100\text{ V} \) and \(E = 10 \times 10^3 \text{ V/m} \). So, \(d = \frac{100}{10 \times 10^3}\).

Step 4: Calculate the Distance

By calculating the expression from the previous step, we find that \(d = \frac{100}{10000} = 0.01\) meters or \(1\) centimeter.

Key Concepts

Equipotential Planes

In the context of electric fields, equipotential planes are imaginary surfaces where the electric potential is the same at every point. These planes are crucial because they simplify the study of electric fields, especially in uniform fields.
Equipotential planes are always perpendicular to the direction of the electric field. Therefore, if you have a uniform electric field pointing vertically upward, the equipotential planes will be horizontal. This perpendicularity ensures no work is required to move a charge along these planes, as the potential energy remains constant.
In practice, these planes can be visualized by imagining layers that never intersect each other. Think of them as flat sheets stacked parallel above one another, with each sheet representing a different potential level.

Electric Potential Difference

The electric potential difference, often referred to simply as voltage, is a measure of the work done to move a unit charge between two points in an electric field. It tells us how much potential energy has changed from one plane to another.
The formula for electric potential difference is

• \( \Delta V = E \times d \)

where \( \Delta V \) is the potential difference, \( E \) is the electric field strength, and \( d \) is the distance between points.
This equation helps determine

• how energy changes with distance in a field,
• and whether the field is capable of doing work on a charge.

In simpler terms, if the voltage difference is larger, more energy is required (or released) to move a charge across those points.

Distance in Electric Fields

In a uniform electric field, the distance between equipotential planes is directly related to the electric potential difference and the strength of the electric field. The previously mentioned formula \( \Delta V = E \times d \) is rearranged to find distance as:

• \( d = \frac{\Delta V}{E} \)

Here, \( d \) is the distance between planes, \( \Delta V \) is the change in potential, and \( E \) is the electric field strength.
This means that as the electric field intensity increases, the distance between equipotential planes decreases for the same potential difference. Vice versa, for a specific field strength, greater potential differences indicate larger stretches between planes.
Understanding how to calculate and interpret these distances is essential in practical applications, like designing electronic circuits and analyzing field distributions.