Kvant Physics Problem 6

Verified: no
Verdicts: SKIP + SKIP
Solve time: 7m14s
Source on kvant.digital

Problem

Find the charge of a grounded metallic sphere of radius $r$, if a point charge $q$ is located at a distance $R$ from its center.

II All-Union Physics Olympiad

Setup and Assumptions

A grounded conducting sphere of radius $r$ is fixed in vacuum. A point charge $q$ is placed at a distance $R$ from the center of the sphere, with $R>r$, measured from the same inertial frame in which the sphere is at rest. The sphere is connected to the Earth, so its electric potential is constrained to zero.

The unknown is the total electric charge $Q$ induced on the sphere. The charge distribution on the sphere is not required explicitly, only its net value.

The conductor is assumed ideal, so electrostatic equilibrium is established instantaneously, the electric field inside the metal vanishes, and charges reside on the surface. Edge effects beyond electrostatics are neglected. The surrounding medium is vacuum with permittivity $\varepsilon_0$.

Physical Principles

Electrostatic equilibrium in a conductor requires the electric field inside the conducting material to vanish and the potential on the surface to be constant.

The boundary condition for a grounded conductor fixes the potential on the surface as $V=0$.

The method of images replaces the conductor with fictitious charges that reproduce the same potential in the region outside the conductor while satisfying the boundary conditions.

The uniqueness theorem for solutions of Laplace’s equation states that if a potential satisfies Poisson’s equation in a region and matches boundary conditions on all surfaces, then it is the unique physical solution in that region.

The total induced charge on a conductor equals the total charge that must be placed on the conductor in the equivalent electrostatic configuration that reproduces the same external field.

Derivation

A point charge $q$ located at distance $R$ from the center of a grounded conducting sphere of radius $r$ can be treated using an image charge $q'$ placed on the line connecting the sphere’s center and the external charge, at a distance $a$ from the center inside the sphere.

The image construction that enforces the boundary condition $V=0$ on the sphere surface gives the standard relations

$$q' = -q \frac{r}{R}, \qquad a = \frac{r^2}{R}.$$

The potential outside the sphere produced by the real charge $q$ and the image charge $q'$ satisfies the condition that the sphere surface is an equipotential at zero. By the uniqueness theorem, this configuration is equivalent to the physical situation with the grounded sphere.

The induced charge on the conductor must reproduce the same external potential as the image charge, since both produce identical fields in the region $r_{\text{space}} \ge r$. Therefore the total induced charge on the sphere equals the image charge:

$$Q = q'.$$

Substituting the expression for the image charge yields

$$Q = -q \frac{r}{R}.$$

Result

The total charge on the grounded sphere is

$$Q = -q \frac{r}{R}.$$

No further numerical values are specified in the problem statement, so the result remains in symbolic form with units of coulombs:

$$Q = -q \frac{r}{R} ;; [\text{C}].$$

Sanity Checks

Dimensional consistency holds since the ratio $r/R$ is dimensionless, so $Q$ has the same units as $q$, namely coulombs.

In the limit $R \to \infty$, the induced charge approaches $Q \to 0$, consistent with the vanishing influence of a distant charge on a grounded conductor.

In the limit $R \to r^{+}$, the induced charge approaches $Q \to -q$, corresponding to strong screening when the external charge approaches the sphere surface.

The sign is consistent with physical intuition: a positive external charge induces a negative net charge on a grounded conductor, ensuring the potential on the sphere remains zero. The most common error in this derivation arises from confusing the image charge location with its magnitude, which would incorrectly modify the factor $r/R$ in the final expression.