A point charge q is placed a distance L from a grounded plate (z=0). Calculate the...

A point charge q is placed a distance L from a grounded plate (z=0). Calculate the charge distribution on the plate.

Expert Solution

The simplest example of method of image charges is that of a point charge, with charge q, located at above an infinite grounded(i.e.: ) conducting plate in the xy-plane. To simplify this problem, we may replace the plate of equipotential with a charge –q, located at . This arrangement will produce the same electric field at any point for which (i.e.: above the conducting plate), and satisfies the boundary condition that the potential along the plate must be zero. This situation is equivalent to the original setup, and so the force on the real charge can now be calculated with Coulomb's law between two point charges.

The potential at any point in space, due to these two point charges of charge +q at +a and -q at -a on the z-axis, is given in cylindrical coordinates as

The surface charge on the grounded plane is therefore given by

In addition, the total charge induced on the conducting plane will be the integral of the charge density over the entire plane, so:

The total charge induced on the plane turns out to be simply –q.

Because electric fields satisfy the superposition principle, a conducting plane below multiple point charges can be replaced by the mirror images of each of the charges individually, with no other modifications necessary.

genius_generous answered 1 hour ago

A point of charge q is placed at the point (-d,0) and a second point charge...

A point of charge q is placed at the point (-d,0) and a second point charge of charge 2q is placed at point (2d,0). a) What is the force on the third charge of charge -q placed at point (0,d) if d=190nm and q=250nC? b) find the two positive charges placed on the x-axis? c) What is the electric field at the origin?

A charge q is a distance b > R from the center of a grounded conducting...

A charge q is a distance b > R from the center of a grounded conducting sphere (that is, a sphere at zero potential, perhaps connected by a thin wire to an enclosing spherical shell very, very far away) of radius R. We want to find an image charge q′ at a distance a < R from the center of the sphere along the same line (take it to be the z-axis) as q such that the potential due to...

There is a line of negative charge (-Q) along the y axis (from 0 to L)....

There is a line of negative charge (-Q) along the y axis (from 0 to L). There is a line of positive charge (+Q) along the x-axis (from 0 to L). A) Find the electric field at the point (L/2, L/2). Sketch the direction of the field at this point on a diagram of the situation. B) Find the potential at the point (L/2, L/2).

A cube with side a is located at the origin,a point charge q is placed at...

A cube with side a is located at the origin,a point charge q is placed at each vertex .find the charge density in term of Dirac delta function

(a) A positive point charge Q is placed at the center of a cube. Draw the...

(a) A positive point charge Q is placed at the center of a cube. Draw the field lines for this charge configuration. Include at least 7 field lines and be sure to show them passing through the surface of the cube. (b) What is the flux through one side of this cube? Hint: Do not under any circumstances integrate to find the flux. Think about how many sides the cube has and whether or not the flux through each face...

A point charge Q is placed at the center of a cube of side L. What is the flux through one face of the cube?

A positive charge q is fixed at the point x = 0, y = 0, and...

A positive charge q is fixed at the point x = 0, y = 0, and a negative charge -2q is fixed at the point x = a, y = 0. a) Derive an expression for the potential V at points on the x -axis as a function of the coordinate x . Take V to be zero at an infinite distance from the charges. (Express your answer in terms of the given quantities and appropriate constants). V(x) = b)...

Three point charges are placed on the y-axis, a charge q at y=a, a charge -2q...

Three point charges are placed on the y-axis, a charge q at y=a, a charge -2q at the origin and a charge q at y=-a. Such an arrangement is called an electric quadrupole. (a) Find the magnitude and direction of the electric field at points on the positive x-axis. (b)What would be the force exerted by these three charges on a fourth charge 2q placed at (b,0). Explain the steps along the way and thought process behind solving.

A positive charge q is fixed at the point x=0,y=0 and a negative charge -2q is...

A positive charge q is fixed at the point x=0,y=0 and a negative charge -2q is fixed at the point x=a,y=0. Part A: Derive an expression for the potential V at points on the y-axis as a function of the coordinate y. Take V to be zero at an infinite distance from the charges. Part B: At which positions on the y-axis is V = 0? Part C: What does the answer to part A become when y>>a? Part D:...

Derive the formula :, where L is distance, m is mass, z is charge, V is...

Derive the formula :, where L is distance, m is mass, z is charge, V is potential

Question