Final answer:
The work required to move a +0.50 µC charge near two +30 µC charges is found by calculating the change in electric potential energy as the charge moves between two points in the electric field created by the charges.
Step-by-step explanation:
To calculate the work required to move a +0.50 µC test charge from a point midway between two identical +30 µC charges to a point 10 cm closer to either charge, we need to use the concept of electric potential energy in the context of electrostatics. The difference in potential energy between two points is equal to the work done in moving a charge from one point to the other in an electric field. The electric potential at a point in space due to a point charge Q at a distance r is given by V = kQ/r, where k is Coulomb's constant (8.9875 x 109 N m2/C2).
First, determine the potentials at the starting and ending points due to each of the 30 µC charges, then find the difference in potential energy for the test charge moving between these two points: ΔPE = q(ΔV), where q is the charge of the test charge and ΔV is the difference in potentials. The initial distance from each 30 µC charge is 16 cm (midway), and the final distance to the closer charge is 16 cm - 10 cm = 6 cm. Then, calculate the potentials and difference, and finally determine the work.