A +30 µC charge is placed 44 cm from an identical +30 µC charge. How much work would be required to move a +0.69 µC test charge from a point midway between them to a point 14 cm closer to either of the charges?
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To calculate the work required to move a test charge in an electric field, we need to multiply the electric force exerted on the charge by the displacement over which it is moved. In this case, the electric force is given by Coulomb's law:
F = k * (|q1| * |q2|) / r^2
where:
F is the electric force,
k is Coulomb's constant (9.0 × 10^9 Nm^2/C^2),
|q1| and |q2| are the magnitudes of the charges, and
r is the separation between the charges.
First, let's calculate the electric force exerted on the test charge when it is midway between the two +30 µC charges. The force between two charges of the same magnitude will be repulsive, so the charges will push away from each other. Since both charges are identical, the electric force on the test charge due to each of them will be the same:
F_midway = k * (|q1| * |q2|) / r^2
= (9.0 × 10^9 Nm^2/C^2) * ((30 × 10^-6 C) * (30 × 10^-6 C)) / (0.44 m)^2
≈ 1.093 N
Next, let's calculate the displacement over which the test charge is moved. The distance from the point midway between the charges to the point 14 cm closer to either of the charges is:
d = 0.44 m - 0.14 m
= 0.30 m
Now, we can calculate the work done:
Work = Force * Displacement
= F_midway * d
≈ 1.093 N * 0.30 m
≈ 0.328 Nm (or Joules)
Therefore, approximately 0.328 Joules of work would be required to move the +0.69 µC test charge from a point midway between the charges to a point 14 cm closer to either of the charges.