A +Q and a +2Q point charges were placed at the vertices of an
equilateral triangle of side a, A third —2Q charge is placed at point P.
What is the work done by an external force in moving —2Q from a position at
infinity to P?
The work done by an external force in moving a point charge from infinity to a point in the field of other charges is equal to the change in its potential energy. The potential energy of a point charge Q at a point in an electric field is given by:
U = QV
where V is the electric potential at that point. The electric potential at a point in the field of other charges is the work done per unit charge by an external force in bringing a small positive test charge from infinity to that point. It is defined as:
V = W/Q
where W is the work done by an external force in bringing the test charge from infinity to that point.
In this problem, we can find the electric potential at point P due to the charges +Q and +2Q by summing the contributions from each charge. The electric potential at a point in the field of a point charge Q is given by:
V_Q = kQ/r
where k is the Coulomb constant and r is the distance between the point charge and the point at which the potential is being calculated.
At point P, the distance between +Q and -2Q is a, and the distance between +2Q and -2Q is 2a. Using the principle of superposition, the electric potential at point P due to the charges +Q and +2Q is:
V_P = kQ/a + k(2Q)/(2a)
= kQ/a + kQ/a
= 2kQ/a
The electric potential at infinity is zero, since there are no charges present at infinity. Therefore, the change in potential energy of the charge -2Q as it is moved from infinity to point P is:
ΔU = QΔV
= -2Q(2kQ/a)
= -4kQ^2/a
Since the charge -2Q is negative, the work done by an external force in moving it from infinity to point P is positive. Therefore, the work done by an external force in moving the charge -2Q from infinity to point P is:
W = -ΔU
= 4kQ^2/a
= (1/4πε₀)(Q^2/a)
where ε₀ is the permittivity of free space.
Therefore, the work done by an external force in moving the charge -2Q from infinity to point P is (1/4πε₀)(Q^2/a).