The Method of Images

The Core Idea

Imagine a point charge \( q \) placed a distance \( d \) above an infinite, grounded conducting plane. The electric field lines from \( q \) terminate perpendicularly on the plane. Instead of solving the differential equation for this setup, we can imagine removing the plane and placing a fictitious charge \( -q \) (the image charge) at a distance \( d \) below the plane, symmetric to the original charge. The potential at any point above the plane due to these two charges is identical to the potential of the original charge-plane system. The plane itself becomes an equipotential surface at zero potential. We do not place the image charge in the physical region; it exists only in the mathematical extension of the space to help us calculate the field in the real region.

Point Charge Near a Grounded Plane

Consider a charge \( q \) at \( (0,0,d) \) and a grounded plane at \( z=0 \). We propose an image charge \( q' = -q \) at \( (0,0,-d) \). The potential at a point \( P(x,y,z) \) with \( z > 0 \) is the superposition of the potentials from \( q \) and \( q' \):

On the plane \( z=0 \), the distances to \( q \) and \( q' \) are equal, so \( \phi(x,y,0) = 0 \). This satisfies the Dirichlet boundary condition. By the Uniqueness Theorem, this is the exact solution for \( z > 0 \). The electric field is obtained by \( \vec{E} = -\nabla\phi \). The induced surface charge density \( \sigma \) on the plane can be found from the discontinuity in the normal component of the electric field: \( \sigma = \varepsilon_0 E_z \) (just above the surface). Integrating \( \sigma \) over the plane yields a total induced charge of \( -q \), consistent with Gauss's law.

Point Charge Near a Grounded Sphere

For a grounded conducting sphere of radius \( a \) with a charge \( q \) located at a distance \( b > a \) from the center, the image charge \( q' \) is not simply \( -q \). To satisfy the boundary condition \( \phi = 0 \) on the sphere's surface, the image charge must have magnitude \( q' = -\frac{a}{b}q \) and be located at a distance \( b' = \frac{a^2}{b} \) from the center, along the line connecting the center to \( q \). This specific placement ensures that the potential vanishes everywhere on the spherical surface. The potential in the region outside the sphere is the sum of the potentials due to \( q \) and \( q' \). The force on the charge \( q \) is calculated by considering the field produced by the image charge \( q' \) at the location of \( q \), since the field of \( q \) itself does not exert a net force on \( q \).

Energy and Work

The electrostatic energy of the system can be calculated in two ways. First, by integrating the energy density \( u = \frac{1}{2}\varepsilon_0 E^2 \) over the region outside the conductor. Second, by considering the work done to bring the charge \( q \) from infinity to its position. The work done is equal to the potential energy \( U = \frac{1}{2} q \phi_{\text{image}}(q) \), where \( \phi_{\text{image}}(q) \) is the potential at the location of \( q \) due to the image charge alone. The factor of \( \frac{1}{2} \) arises because the image charge is a mathematical construct that moves as the real charge moves; the physical work is half the product of the charge and the potential due to the induced charges. This energy is always negative for a grounded conductor, indicating that the charge is attracted to the conductor.