Problem 29 - University Physics with Modern Physics

Question

A uniform electric field has magnitude \(E\) and is directed in the negative \(x\) -direction. The potential difference between point \(a\) (at \(x=0.60 \mathrm{m}\) ) and point \(b\) (at \(x=0.90 \mathrm{m} )\) is 240 \(\mathrm{V}\) . (a) Which point, \(a\) or \(b\) , is at the higher potential? (b) Calculate the value of E. (c) A negative point charge \(q=-0.200 \mu \mathrm{C}\) is moved from \(b\) to a. Calculate the work done on the point charge by the electric field.

Step-by-Step Solution

Answer

a) Point a is at a higher potential. b) The electric field E is 800 V/m. c) The work done is -48 \(\mu J\).

Step 1: Understand Electric Potential

The electric potential decreases in the direction of the electric field. Since the electric field is directed in the negative x-direction, the potential decreases from point a to point b.

Step 2: Determine Higher Potential Point

Since the electric potential decreases in the direction of the field, point a (at x = 0.60 m) is at a higher potential than point b (at x = 0.90 m).

Step 3: Use Potential Difference Formula

The potential difference between two points is given by the formula: \( V = E \cdot d \), where \( d \) is the distance between the points. With a potential difference of 240 V and a distance of 0.90 m - 0.60 m = 0.30 m, we can find E.

Step 4: Calculate E

Rearrange the potential difference formula to find \( E \): \( E = \frac{V}{d} = \frac{240 \text{ V}}{0.30 \text{ m}} = 800 \text{ V/m} \).

Step 5: Calculate Work Done

The work done on a charge moved in an electric field is given by \( W = q \cdot V \), where \( q = -0.200 \mu C = -0.200 \times 10^{-6} C \) and \( V = 240 \text{ V} \). Hence, the work done is \( W = (-0.200 \times 10^{-6} C) \cdot 240 \text{ V} = -48 \times 10^{-6} J = -48 \mu J \).

Key Concepts

Electric Potential

Electric potential is similar to gravitational potential energy. It is the potential energy stored due to an object's position within an electric field. Think of it as the energy that influences how a charged particle will move. When a test charge (positive or negative) is placed in an electric field, it has a certain potential energy depending on its location within that field.

The potential energy per unit charge is called electric potential, often denoted as "V" and measured in volts (V). Here is the formula to compute it:

• Electric Potential (V) = Electric Potential Energy / Charge

Understanding this interaction helps predict how a charge will behave when it's released in the electric field. In scenarios where a uniform electric field is involved, realizing that the potential decreases in the direction of the field’s force gives insight into the forces acting upon the charge.

Potential Difference

Potential difference is the change in electric potential energy between two points in a field. It tells us how much energy per charge we will gain or lose when moved between two points. The difference is measured in volts and is often referred to as "voltage."

To find the potential difference, we often use the concept that the electric field will act along a path, causing the potential to change. In a uniform electric field, this can be calculated using:

• Potential Difference (V) = Electric Field (E) x Distance (d)

In our exercise, the potential difference of 240 V indicates that point "a," being opposite the field direction, has a higher potential compared to point "b." Knowing the potential difference allows one to calculate the magnitude of the electric field, reinforcing the relationship that a greater difference will indicate a stronger field over a given distance.

Work Done in Electric Field

When a charge moves through an electric field, work is done on the charge or by the charge. This concept is crucial in understanding energy transfer within electric systems. The work done when moving a charge between two points is calculated via:

• Work (W) = Charge (q) x Potential Difference (V)

In the context of our exercise, a negative charge is moved, and evaluating the work done involves simply multiplying the charge and the potential difference. This multiplication gives insight into how much energy is required to move that charge within the electric field.

Remember that if the work done is negative, as in our problem, it indicates that energy is being released by the charge while moving within the field's influence, as it naturally moves towards lower potential energy locations. Understanding this helps comprehend the energetic interactions and transitions in electrical contexts.