Question

A point charge \(Q\) is placed a distance \(R\) from the center of a conducting sphere of radius \(a\), with \(R>a\) (the point charge is outside the sphere). The sphere is grounded, that is, connected to a distant, unlimited source and/or sink of charge at zero potential. (Neither the distant ground nor the connection directly affects the electric field in the vicinity of the charge and sphere.) As a result, the sphere acquires a charge opposite in sign to \(Q\), and the point charge experiences an attractive force toward the sphere. a) Remarkably, the electric field outside the sphere is the same as would be produced by the point charge \(Q\) plus an imaginary mirror-image point charge \(q\), with magnitude and location that make the set of points corresponding to the surface of the sphere an equipotential of potential zero. That is, the imaginary point charge produces the same field contribution outside the sphere as the actual surface charge on the sphere. Calculate the value and location of \(q\). (Hint: By symmetry, \(q\) must lie somewhere on the axis that passes through the center of the sphere and the location of \(Q .)\) b) Calculate the force exerted on point charge \(Q\) and directed toward the sphere, in terms of the original quantities \(Q, R\), and \(a\). c) Determine the actual nonuniform surface charge distribution on the conducting sphere.

Short answer

Answer: The value of the image charge q is given by \(q = -\frac{a}{R-a} \cdot Q\), and its location is given by \(\vec{r}_q = \frac{a^2}{R} \cdot \hat{r}\), where a is the sphere's radius, R is the distance between point charge Q and the sphere's center, and \(\hat{r}\) is the unit vector in the direction from the sphere's center to point charge Q.

Step-by-step solution

Sketch the problem

First, let's make a sketch representing the point charge \(Q\), grounded conducting sphere with radius \(a\), and the distance \(R\) between them. Also, locate the mirror-image point charge \(q\) on the axis passing through the center of the sphere.

Find the electric potential produced by point charge Q

The electric potential produced by point charge Q at a position \(\vec{r}\) is given by: \[ V_{Q} (\vec{r}) = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{|\vec{r} - \vec{r}_Q|} \] Where \(\epsilon_0\) is the vacuum permittivity and \(\vec{r}_Q\) is the position vector of point charge Q.

Find the electric potential produced by point charge q

The electric potential produced by point charge q at a position \(\vec{r}\) is given by: \[ V_{q} (\vec{r}) = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q}{|\vec{r} - \vec{r}_q|} \] Where \(\vec{r}_q\) is the position vector of point charge q.

Find the equipotential for V=0

The total electric potential at the surface of the sphere will be the sum of the potentials created by Q and q. Since the sphere is grounded, it is at zero potential. So, we have: \[ V_{Q}(\vec{r}_{surface}) + V_{q}(\vec{r}_{surface}) = 0 \] Where \(\vec{r}_{surface}\) is the position vector of any point on the surface of the sphere.

Calculate the value and location of q based on V=0

Solve the equation obtained in step 4 for the value of q and its position vector \(\vec{r}_q\). This can be done by analyzing where the equipotential of potential zero lies. After calculations, we will find that: \[ q = -\frac{a}{R-a} \cdot Q \] and \[ \vec{r}_q = \frac{a^2}{R} \cdot \hat{r} \] Where \(\hat{r}\) is the unit vector in the direction from sphere's center to point charge Q.

Determine the force on point charge Q

The force on point charge Q is the force between the original charge Q and the mirror-image charge q. We can calculate the force using Coulomb's law: \[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q \cdot q}{|\vec{r}_Q - \vec{r}_q|^2} \cdot \hat{r} \] Substitute the values of q and \(\vec{r}_q\) obtained in step 5.

Determine the surface charge distribution on the conducting sphere

Because of the spherical symmetry, we can determine the surface charge density using Gauss's law: \[ \sigma (\theta) = - \frac{1}{2 \pi a^2} \cdot \frac{Q}{R} \cdot (\frac{a}{R-a})^3 \cdot \frac{1}{\sin \theta} \] Where \(\theta\) is the polar angle measured from the axis going through the center of the sphere and the location of point charge Q.