What is the distribution of induced charge on a conducting sphere of radius *R* placed in a uniform electric field **E**_{0}?

Let us place the origin at the center of the sphere, and choose the positive *z* axis in the direction of **E**_{0}. Now, consider the sphere without the uniform field, and instead, with a charge of +*Q* placed on the *z* axis at *z*=-*a* and a charge of –*Q* placed at *z*=+*a*, with *a*≫*R*. Then, as noted here, the field near the origin is , and if we take *a*,*Q*→∞ with held constant, the result approaches a uniform electric field. Thus, we choose that *Q*=2π*ε*_{0}*E*_{0}*a*^{2} (*E*_{0}=|**E**_{0}|), then the limit as *a*→∞ is the desired uniform field.

Now, as in this post, we use the method of image charges. The point charge –*Q* at *z*=+*a* has an image charge of at . Similarly, the +*Q* charge at *z*=-*a* has an image charge of at . The distance between these charges is , and the product of this distance and charge of the image points is

. As *a*→∞, *d*→0, and we see from here that the dipole moment **p**=4π*ε*_{0}*R*^{3}**E**_{0} is held constant in our limit. Thus, in our limit, the image charges approach the point dipole of dipole moment **p**=4π*ε*_{0}*R*^{3}**E**_{0}.

The potential for a uniform field is *φ*=-*E*_{0}*z*. The potential for a point dipole is

, which, in this case, with *θ* the angle from the positive z-axis (as usual), then , and

Thus the total potential outside the conducting sphere is

.

And we see by inspection that the potential is constant (specifically, zero) when *r*=*R*, the surface of the sphere, as required.

Now, the outward surface normal for the surface of our conductor is just the unit radial vector, as our conductor is a sphere, and so the directional derivative of the potential along that normal is just . Using the results of this post, we see the surface charge density is

,

which is independent of the radius of the sphere, and which we see integrates over the surface to give a net induced charge of zero.

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Tags: Dipole, Dipole Moment, Electrostatics, Friday Physics, Induced Charge, Method of Images, physics, Surface Charge

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