A +30 µC point charge is placed 32 cm from an identical +30 µC charge. How much work would be required to move a +0.45 µC test charge from a point midway between them to a point 11 cm closer to either of the charges?
To calculate the work required to move the test charge, we need to consider the electric potential energy.
The electric potential energy (U) is given by the equation:
U = k * (q1 * q2) / r
where:
- k is the Coulomb's constant, approximately 8.99 x 10^9 N•m^2/C^2
- q1 and q2 are the charges, in this case, the test charge (+0.45 µC) and the point charge (+30 µC)
- r is the distance between them
In this equation, U represents the electric potential energy at a specific location. The work done on a test charge is equal to the change in electric potential energy (ΔU) when it moves from one location to another.
Now, let's calculate the initial electric potential energy at the first point:
U1 = k * (q1 * q2) / r1
where:
- r1 is the distance between the test charge and the point charge at the midpoint
Next, we calculate the final electric potential energy at the second point:
U2 = k * (q1 * q2) / r2
where:
- r2 is the distance between the test charge and either of the point charges at the closer point
Finally, we can calculate the work done to move the test charge as the difference in potential energy:
Work = U2 - U1
Let's plug in the values and calculate the work!
First, convert the charge values to coulombs:
Test charge (q1) = +0.45 µC = +0.45 x 10^-6 C
Point charges (q2) = +30 µC = +30 x 10^-6 C
Next, calculate the initial and final distances in meters:
Initial distance (r1) = halfway point = 32 cm / 2 = 16 cm = 16 x 10^-2 m
Final distance (r2) = intermediate point = distance r1 - 11 cm = 16 cm - 11 cm = 5 cm = 5 x 10^-2 m
Now, let's calculate the initial and final electric potential energies:
U1 = (8.99 x 10^9 N•m^2/C^2) * (+0.45 x 10^-6 C) * (+30 x 10^-6 C) / (16 x 10^-2 m)
U2 = (8.99 x 10^9 N•m^2/C^2) * (+0.45 x 10^-6 C) * (+30 x 10^-6 C) / (5 x 10^-2 m)
Finally, calculate the work done:
Work = U2 - U1
By plugging in the values into the equations and evaluating the expression, you will find the amount of work required to move the test charge.