The mirror method is to replace the given system to be solved with a suitably configured charge system.
According to the uniqueness theorem, as long as the newly added charge (mirror image charge) does not enter the original solution area to ensure that the charge distribution in the original solution area is not changed, the potential equation in the original solution area can be guaranteed not to be changed, and the boundary of the original area conditions remain unchanged to meet the requirements.
The mirror image method actually replaces the influence of the original boundary on the system with the charge added outside the solution area. Therefore, when using the mirror image method, the original boundary is removed first, and then some charges are appropriately added outside the solution area to adjust The amount of charge, the amount of charge, and the position of the charge make the conditions at the original boundary unchanged. In this way, according to the uniqueness theorem, the solution of the new system is the solution of the original system in the original solution area.
Table of contents
1. A system in which a point charge is placed above an infinitely large grounded conductor plate
2. A system in which a point charge is placed outside a grounded conductor sphere
3. A system in which a point charge is placed outside an ungrounded and uncharged conductor sphere
1. A system in which a point charge is placed above an infinitely large grounded conductor plate
If there is no conductor plate, the electric field is generated by q, and there is no equipotential surface at the conductor plate position. Because of the existence of the conductor plate, the position is forced to become an equipotential surface
The effect of the conductor plate on the electric field is due to the induced charge on the surface. The distribution of the electric force line of the original system is:
The distribution of electric force lines after adding image charge is:
It can be seen that in the upper half space, the potential distribution does not change, the boundary conditions of the original solution area are still satisfied, and the position of the conductor plate is still the equipotential surface, so now it is necessary to solve the position and size of -q
Before adding image charge
For the upper half space:
For the lower half space:
Add image charge
(1) Because the image charge cannot affect the potential distribution of the original solution area, the image charge can only be placed in the lower half space
(2) The magnitude of the induced charge on the conductor plate is q, and the sign is negative, so the size of the image charge is q, and the sign is negative
(3) The electric field generated by the charge q and the image charge -q satisfies:
Therefore, the position of the image charge is -d, and the connection line with q is perpendicular to the conductor plate
(So far, we have determined the size, position, sign, etc. of the image charge. We can use the image charge and point charge system to solve the potential distribution of the upper half space, but it should be noted that for the lower half space, the image charge and point charge system cannot be used solve)
2. A system in which a point charge is placed outside a grounded conductor sphere
A grounded conductive spherical shell has a charge q at a distance d, which determines the image charge
Next to q, the position of the conductor spherical shell is the equipotential surface, and after it is grounded, φ=0
Now determine the size and position of the image charge:
(1) According to symmetry, the image charge must be on the connection line between o and q
(2) Assuming that the image charge distance from point o is b, then the size of b is required, and the size of -q'
After removing the grounded earth’s crust, the system consists of image charges and point charges, and the boundary condition φ=0 must be satisfied, namely:
(The potential at any point on the spherical shell is superimposed by the potential generated by the point charge and the potential generated by the image charge, r and r' are calculated according to the law of cosines)
Organized to get:
Because the potential is equal to zero for any point on the spherical shell, the above formula is always established for any θ, that is:
When q' and b take the above values, the boundary condition is satisfied, that is, the potential distribution in the outer space of the conductor sphere does not change after adding the image charge; then according to the uniqueness theorem, solving the problem of the potential distribution in the outer sphere can be transformed into, by Point charges and image charges form a system to solve potential distribution problems.
in conclusion:
The conductor plate is like a plane mirror, the image charges are equal in size and distance;
The earth shell of the conductor is like a concave-convex mirror, and its size and distance are stretched proportionally.
3. A system in which a point charge is placed outside an ungrounded and uncharged conductor sphere
When the conductive spherical shell is not grounded, the surface will generate induced charges. From the above, the amount of induced charge is -q'; the conductive spherical shell is an equipotential surface, but the potential is not necessarily equal to 0, but a constant. The potential everywhere is equal to the surface potential of the ball.
At present, the system is composed of the induced charge -q' on the surface of the spherical shell and the point charge q, and the image charge is now determined:
(1) The surface of the spherical shell is an equipotential surface, so the image charge must be at the center of the sphere
(2) The size of the image charge can be calculated from the above conclusion
Right now:
Then the original system is transformed into: a charge system composed of q, -q', q'
Inside a conductor sphere:
The potential distribution depends only on -q', q'
Outside the conductor sphere:
Since the q' electric field cannot pass through the sphere, the external electric field depends only on -q', q
On the surface of a conductor sphere:
Potential is constant