A charge +q is placed a distance d above a grounded, infinite conducting plane. In the method of images, what replaces the conductor to produce the same boundary conditions?
AA continuous surface charge distribution spread evenly across the plane
BA single image charge −q placed at the mirror-image position below the plane
CA charge +q placed at the mirror-image position below the plane
DThe boundary conditions are removed, and Laplace's equation is solved numerically
The image charge −q placed at the mirror position below the plane ensures that the combined potential of +q and −q is exactly zero on the grounded plane (by symmetry, the plane is equidistant from both). This satisfies the boundary condition. A +q image would not cancel the potential on the plane; a surface distribution is what we are replacing, not recreating.
Question 2 Multiple Choice
A student uses the image charge −q below the plane to compute the electric field everywhere above the grounded conducting plane. The student's advisor says: 'The image charge doesn't physically exist.' Which statement best resolves this?
AThe advisor is wrong — the image charge is physically real and causes the induced surface charges
BThe advisor is right — the image charge is a mathematical fiction that lives in the conductor region; its field above the plane is real and equals the field from the actual induced surface charges
CThe advisor is right, and the field from the image charge should not be included in the solution above the plane
DThe image charge is real only at the conducting surface, not below it
The image charge is a mathematical device — it lives in the region (inside the conductor) where we are *not* solving for the field. But the uniqueness theorem guarantees that any configuration satisfying the correct boundary conditions and source distribution is the unique solution in the region of interest. The field from the image charge above the plane is physically real and identical to the field the actual induced surface charges would produce.
Question 3 True / False
The force on a charge +q placed a distance d above a grounded conducting plane equals the Coulomb force between +q and its image charge −q at distance 2d away.
TTrue
FFalse
Answer: True
The method of images gives the force on the real charge directly: it is the Coulomb attraction between +q and −q separated by 2d (the real charge at distance d above, image at distance d below). This 'image force' explains why charged particles are attracted to nearby conductors even when the conductor carries no net charge — the conductor rearranges its surface charges to create exactly the response that an opposite image charge would.
Question 4 True / False
The method of images works by directly computing the distribution of induced surface charges on the conductor, then using those charges to find the field.
TTrue
FFalse
Answer: False
This is precisely what the method avoids. Computing induced surface charges directly requires solving an integral equation — the hard problem. Instead, the method asks: is there a simple arrangement of point charges (image charges) outside the domain that produces the correct boundary conditions? If yes, the uniqueness theorem guarantees that the total field in the region of interest is the unique correct solution, without ever computing the surface charge distribution explicitly.
Question 5 Short Answer
Why does the uniqueness theorem guarantee that the image charge solution is correct, even though the image charge is not physically present in the conductor?
Think about your answer, then reveal below.
Model answer: The uniqueness theorem for electrostatics states that if you find any solution to Laplace's equation in a region that satisfies the correct boundary conditions and has the correct source charges, it is the only solution. The image charge configuration satisfies both: the real charge +q is the correct source in the region above the plane, and the combined potential of +q and its image −q equals zero on the grounded plane (correct boundary condition). Therefore, this must be the unique correct solution above the plane — regardless of whether the image charge is real.
Uniqueness is the linchpin of the entire technique. Without it, finding one configuration that satisfies the boundary conditions would not be enough — there could be infinitely many solutions. Uniqueness collapses that possibility: one valid solution is the only solution. The image charge is a clever way to construct that one valid solution without solving the hard integral equation for the surface charge distribution.