Electric charge conservation : When a point charge q is abovean infinite conducting plane, the induced opposite charge will bedistributed on the conducting surface, and the magnitude of theimage charge should be equal to the totalinduced charge.We can also provethe claim by making use of the relationshipbetween the density of the charge andthe electricfieldintensity orthe derivative ofthe electric potentialon the conducting surfaceThe above equivalence is established only for the upper half-spacefor which the source and the boundary condition are both unchanged
Electric charge conservation ：When a point charge q is above an infinite conducting plane, the induced opposite charge will be distributed on the conducting surface, and the magnitude of the image charge should be equal to the total induced charge. The above equivalence is established only for the upper half-space for which the source and the boundary condition are both unchanged. We can also prove the claim by making use of the relationship between the density of the charge and the electric field intensity or the derivative of the electric potential on the conducting surface
For the semi-infinite wedge conducting boundary, the method ofimages is also applicable. However, the images can be found only forconducting wedges with angle given by where n is an integer. Inorder to keep the wedge boundary at zero-potential, several imagecharges arerequired.④q元/3e+When an infinite line charge is nearby an infinite conductingplane, the method of images can be applied as well, based on theprincipleofsuperpositionUV
q For the semi-infinite wedge conducting boundary, the method of images is also applicable. However, the images can be found only for conducting wedges with angle given by where n is an integer. In order to keep the wedge boundary at zero-potential, several image charges are required. 3 π /3 /3 q When an infinite line charge is nearby an infinite conducting plane, the method of images can be applied as well, based on the principle of superposition
(2)Apoint chargeanda conducting sphereTo replace the effect of theboundary of the conducting sphere,let an image point charge q' beplaced on the line segment between9the point charge q and the center ofthe sphere. Then the electric poten-tialon the surface of the sphere isthen givenbyq@4元8r4元rRequiringthat the electric potential at any point on the surfaceof the spherebe zero,theimage charge must beU
f q O （2）A point charge and a conducting sphere. To replace the effect of the boundary of the conducting sphere, let an image point charge q' be placed on the line segment between the point charge q and the center of the sphere. Then the electric potential on the surface of the sphere is then given by r q r q = + 4π 4π Requiring that the electric potential at any point on the surface of the sphere be zero, the image charge must be q r r q = − P a d r q r
The ratiomust be constant for any point on the surface ofthe sphere to obtain an image charge with a fixed value.RIf△OPq'~△OqP,thenThus the quantityof theimage charge should beThe distance d isqadfThe electric fieldintensity outside the sphere can be foundout fromg and qUV
The ratio must be constant for any point on the surface of the sphere to obtain an image charge with a fixed value. r r q f a q = − The distance d is f a d 2 = The electric field intensity outside the sphere can be found out from q and q' . If △OPq' ~ △OqP , then . Thus the quantity of the image charge should be f a r r = f q O P a d r q r
If the conducting sphereis ungrounded, then the opposite chargeswill be induced on the side of the conducting sphere facing the pointcharge, while the induced charge on the other side of the sphereispositive. The totalinduced charge on the surface of the conductingsphere should be zeroIf the image charge q'is put in, then another image charge qis needed in order to satisfythe neutralityconditionq"=-q'The image charge must be atthe center of the sphereto ensurethat the surface of the sphere is an equipotentialsurface.In fact, since the sphereis ungrounded, the electric potentialisnon-zero. Since q and q' produce a zero potential on the surface of thesphere, the secondimage charge q"is presentto produce a certainelectric potential.V
If the conducting sphere is ungrounded, then the opposite charges will be induced on the side of the conducting sphere facing the point charge, while the induced charge on the other side of the sphere is positive. The total induced charge on the surface of the conducting sphere should be zero. The image charge must be at the center of the sphere to ensure that the surface of the sphere is an equipotentialsurface. q = −q If the image charge q' is put in, then another image charge q" is needed in order to satisfy the neutrality condition. In fact, since the sphere is ungrounded, the electric potential is non-zero. Since q and q' produce a zero potential on the surface of the sphere, the second image charge q " is present to produce a certain electric potential