Problem 6 - Applied Physics

Question

An electric field of magnitude \(0.370 \mathrm{~N} / \mathrm{C}\) exerts a force of \(6.20 \times 10^{-4} \mathrm{~N}\) on a test charge placed in the field. What is the magnitude of the test charge?

Step-by-Step Solution

Answer

The magnitude of the test charge is approximately \( 1.68 \times 10^{-3} \mathrm{~C} \).

Step 1: Understand the Relationship

The force exerted by an electric field on a test charge is given by the equation: \( F = qE \), where \( F \) is the force, \( q \) is the charge, and \( E \) is the electric field strength.

Step 2: Identify Given Values

From the problem, we have the electric field magnitude, \( E = 0.370 \mathrm{~N/C} \) and the force exerted, \( F = 6.20 \times 10^{-4} \mathrm{~N} \).

Step 3: Rearrange the Formula

We want to find the charge \( q \). Rearrange the formula \( F = qE \) to solve for \( q \): \( q = \frac{F}{E} \).

Step 4: Substitute Values

Substitute the given values into the equation: \( q = \frac{6.20 \times 10^{-4} \mathrm{~N}}{0.370 \mathrm{~N/C}} \).

Step 5: Calculate the Charge

Perform the calculation: \( q = \frac{6.20 \times 10^{-4}}{0.370} \approx 1.68 \times 10^{-3} \mathrm{~C} \). The charge is \( 1.68 \times 10^{-3} \mathrm{~C} \).

Key Concepts

Test Charge

In the study of electric fields, a test charge is an essential concept. A test charge is often a small amount of electric charge placed in an electric field to experience the field's effects. It serves as a probe to measure or observe the behavior of the field.

• Characteristics: It is typically considered positive and small enough to not disturb the field itself. The goal is to keep the electric field's original conditions unchanged by the test charge's presence.
• Purpose: By observing the force that the field exerts on the test charge, we can determine characteristics like the magnitude and direction of the field.

Understanding test charges helps us conceptualize how fields operate and interact with point charges.

Force Calculation

Determining the force experienced by a test charge within an electric field involves simple arithmetic based on a fundamental formula. This concept is crucial because it allows us to quantify the interaction between electric fields and charges.
The relationship is given by the formula: \[ F = qE \] where:

• \( F \) is the force exerted on the charge, measured in newtons (N).
• \( q \) is the magnitude of the charge, measured in coulombs (C).
• \( E \) is the electric field strength, measured in newtons per coulomb (N/C).

This equation means that the force is directly proportional to both the charge and the electric field's strength. In practice, this allows for the calculation of any of these three values when the other two are known, providing us tools to measure real-world electric interactions.

Electric Field Strength

Electric field strength is a measure of how an electric field influences the space around it and its capacity to exert force on electric charges.

• Definition: It is defined as the force per unit charge exerted by an electric field on a positive test charge placed in the field.
• Formula: As used, the formula \( E = \frac{F}{q} \) helps in understanding how strong the electric field is by illustrating the relationship between force and charge.
• Units: The unit of electric field strength is newtons per coulomb (N/C).

Understanding electric field strength helps us predict how strong fields will affect any charges within range. By knowing the field strength, we can also anticipate how the field will change around different charges or configurations of electric elements.

Charge Determination

Charge determination is the process of calculating the amount of electric charge using known values of force and electric field strength. This calculation is critical in both practical applications and theoretical physics.
To determine the charge magnitude, a rearranged version of the basic force equation is used:
\[ q = \frac{F}{E} \]

• Identify Components: First, identify the given electric field strength \(E\) and the force \(F\).
• Substitution: Plug these values into the rearranged formula.
• Calculation: Perform the arithmetic to solve for \(q\), the charge magnitude.

In our specific scenario, calculating the charge allows us to find out exactly how much charge is on the test charge. This value can be used for further applications and helps in better understanding the interactions within the field.