A +35 µC charge is placed 42 cm from an identical +35 µC charge. How much work would be required to move a +0.55 µC test charge from a point midway between them to a point 8 cm closer to either of the charges?
To find the work required to move the test charge, we can use the equation:
Work = Change in Potential Energy
The potential energy between two charges in an electric field can be calculated using the formula:
Potential Energy = (k * q1 * q2) / r
Where:
k is the electrostatic constant (k = 8.99 * 10^9 N m^2/C^2)
q1 and q2 are the charges
r is the distance between the charges
Let's break down the calculation step by step:
Step 1: Calculate the initial potential energy at the starting point.
Given that the test charge is placed midway between the two charges, the distance from each charge to the test charge is half the original distance. Therefore, the initial distance (r1) is 42 cm / 2 = 21 cm = 0.21 m.
Potential Energy1 = (k * q1 * q_test) / r1
Step 2: Calculate the final potential energy at the ending point.
To find the final distance (r2), we take the initial distance and subtract 8 cm. Therefore, r2 = r1 - 0.08 m.
Potential Energy2 = (k * q1 * q_test) / r2
Step 3: Calculate the change in potential energy.
Change in Potential Energy = Potential Energy2 - Potential Energy1
Step 4: Calculate the work required using the equation: Work = Change in Potential Energy
Let's plug in the given values and calculate the work:
q1 = q_test = +35 µC = 35 * 10^-6 C
k = 8.99 * 10^9 N m^2 / C^2
r1 = 0.21 m
r2 = r1 - 0.08 m
Potential Energy1 = (8.99 * 10^9 * 35 * 10^-6 * 35 * 10^-6) / 0.21
Potential Energy2 = (8.99 * 10^9 * 35 * 10^-6 * 35 * 10^-6) / (0.21 - 0.08)
Change in Potential Energy = Potential Energy2 - Potential Energy1
Work = Change in Potential Energy
Now, let's calculate the values step by step:
Potential Energy1 = (8.99 * 10^9 * 35 * 10^-6 * 35 * 10^-6) / 0.21
= 0.0093 J
Potential Energy2 = (8.99 * 10^9 * 35 * 10^-6 * 35 * 10^-6) / (0.21 - 0.08)
= 0.017 J
Change in Potential Energy = Potential Energy2 - Potential Energy1
= 0.017 J - 0.0093 J
= 0.0077 J
Therefore, the work required to move the +0.55 µC test charge from a point midway between them to a point 8 cm closer to either of the charges is approximately 0.0077 Joules.
To calculate the work required to move a test charge between two charges, we need to consider the electrostatic potential energy.
The formula for electrostatic potential energy is given by:
U = k * (q1 * q2) / r
Where U is the potential energy, k is the electrostatic constant (k = 8.99 * 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
In this case, the two charges are both +35 µC (35 * 10^(-6) C) and the distance between them is 42 cm (0.42 m). Our test charge is +0.55 µC (0.55 * 10^(-6) C) and we want to move it from a point midway between the charges to a point 8 cm closer to either of the charges.
First, let's calculate the initial potential energy (U_initial) when the test charge is at the midpoint between the charges.
U_initial = k * (q1 * q2) / r_initial
Where r_initial is the initial distance between the test charge and the charges, which is half of the distance between the two charges.
r_initial = r / 2 = 0.21 m
U_initial = (8.99 * 10^9 N m^2/C^2) * ((35 * 10^(-6) C)^2) / 0.21 m
Next, let's calculate the final potential energy (U_final) when the test charge is at a point 8 cm closer to either of the charges.
r_final = r_initial - 0.08 m
U_final = (8.99 * 10^9 N m^2/C^2) * ((35 * 10^(-6) C)^2) / r_final
Finally, let's calculate the work required to move the test charge from the initial point to the final point.
Work = U_initial - U_final
Substitute the values into the formulas and calculate to find the final answer.