Problem 25 - 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 \(q=-0.200 \mu C\) 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 higher potential. (b) \( E = 800 \) V/m. (c) Work done is 48 \( \mu \)J.

Step 1: Understanding Electric Potential Difference

The potential difference between two points is given by \( V_b - V_a = -E \cdot d \), where \( d \) is the distance between the points and \( E \) is the electric field. Here, the potential difference \( V_a - V_b = 240 \) V is given, which means \( V_a > V_b \), as a positive potential difference indicates point \(a\) is at a higher potential.

Step 2: Calculate the Electric Field Strength

From the potential difference equation \( V_a - V_b = -E \cdot d \), rearrange to find \( E = \frac{V_a - V_b}{-d} \). Substituting values, \( E = \frac{240}{-(0.90 - 0.60)} = \frac{240}{-0.30} = -800 \) V/m. The negative sign confirms the field is in the negative \(x\)-direction.

Step 3: Calculate Work Done by the Electric Field

Work done, \( W \), by the electric field on a charge moving between two points is given by \( W = q(V_b - V_a) \). Substituting \( q = -0.200 \times 10^{-6} \) C and \( V_b - V_a = -240 \) V, we have \( W = (-0.200 \times 10^{-6}) \times (-240) = 48 \times 10^{-6} \) J or \( 48 \mu \)J.

Key Concepts

Potential Difference

Potential difference is the measure of the energy difference per unit charge between two points in an electric field. It tells us how much work is needed to move a charge between these points. In the context of this problem, the potential difference between points \( a \) and \( b \) is given as 240 \( \mathrm{V} \). This indicates that point \( a \) is at a higher potential than point \( b \), since the potential difference is positive when calculated from \( a \) to \( b \).
For example, if you imagine a battery where one side is higher than the other, the side with more energy per electron is at a higher potential.
This concept is crucial because it allows us to calculate other aspects of the electric field like strength and work done.

Electric Potential

Electric potential at a point is defined as the work done in bringing a unit positive charge from infinity to that point, without any acceleration. It's a scalar quantity measured in volts (\( \mathrm{V} \)).
In a uniform electric field, the potential decreases linearly with distance in the field's direction. This means the farther you move against the field lines, the more work you have to put in, making the potential higher.

• Higher potential: Point where you have to push more against the electric field.
• Lower potential: Point where it becomes easier, requiring less push or work against the field.

This understanding helps connect the potential difference and electric field strength.

Work Done by Electric Field

The work done by an electric field when moving a charge from one point to another is calculated using the formula \( W = q(V_b - V_a) \). It is essentially the energy required to move a charge between two points in an electric field.
In this problem, moving a negative point charge of \( q = -0.200 \mu \mathrm{C} \) from point \( b \) to point \( a \) implies the electric field is doing work on the charge. Substitution of values shows the work done is \( 48 \mu \mathrm{J} \).
It is important to remember:

• Work done is positive if the field assists in moving the charge.
• Work done is negative if you have to do the work against the field.

This concept provides insight into how energy is transferred in an electric field.

Uniform Electric Field

A uniform electric field has a constant magnitude and direction. When depicted in a diagram, the field lines are parallel and evenly spaced throughout the region.
In this problem, the field is directed in the negative \( x \)-direction. This setup means the same amount of force and energy change are experienced at any point in the field when moving a charge.

• Fields like these simplify calculations since field strength and direction don't vary.
• Potential difference across any two points is directly proportional to the distance between them.

Understanding a uniform electric field helps predict and calculate how charges behave within this field smoothly.

Point Charge

A point charge is a hypothetical charge located at a singular point in space. It makes calculations simpler by assuming the charge's size or shape doesn't affect the overall analysis.
This type of charge is pivotal in electrostatics since it allows us to conceptualize how electric fields interact with a charged particle.

• For our problem, a negative point charge \( -0.200 \mu \mathrm{C} \) is moved, highlighting how it experiences the electric field.
• Makes it easier to calculate forces, work done, and potential changes due to its independent nature.

Understanding the characteristics of a point charge enhances our ability to model and predict physical phenomena in electric fields.