Surround Ground

Electricity and Magnetism Level 3

A $1~\mbox{nC}$ charge is placed inside a grounded conducting hollow sphere of radius $10~\mbox{cm}$ centered on the origin. If the charge is placed at $(5~\mbox{cm},0,0)$ , what is the electric potential at $(-5~\mbox{cm},0,0)$ ?

Details and assumptions

Let $1/4 \pi \epsilon_0=K=9\times 10^9 ~\mbox{Nm}^2/\mbox{C}^2$ .

The answer is 18.

1 solution

Vitor Terra
Dec 28, 2013

By the method of images, the potential produced by the induced charges on the sphere is equivalent to the potential produced by a -2 nC charge placed at (20 cm, 0, 0). In other words, the potential produced by the 1 nC charge and the grounded sphere is equivalent to the potential produced by both the 1 nC and -2 nC charges. You can verify this by evaluating the potential created by these charges at the points where the surface of the sphere would be (in other words, points distant 10 cm from the origin) - it will be zero, which makes sense, since the sphere in the original system was grounded.

Ok, the potential at the point (-5 cm, 0, 0) can be calculated using the equivalent configuration of charges: 1nC at (5 cm, 0, 0) and -2 nC at (20 cm, 0, 0):

$V = \frac{Kq_{1}}{r_{1}} + \frac{Kq_{2}}{r_{2}} = \frac{9 \cdot 10^9 \cdot 10^{-9}}{0.1} + \frac{9 \cdot 10^9 \cdot (-2 \cdot 10^{-9})}{0.25} = 90 - 72 = 18V$

"But how did you manage to find the right magnitude and position of the auxiliary charge?" There's an explanation here or in Introduction to Electrodynamics by D. J. Griffiths, page 124.

What is the concept behind imaging of charge?????????? Can't understand how you found the magnitude of the charge to be placed at (20cm ,0 ,0).

Harshal Sharma - 7 years, 5 months ago

U need to learn about the Apollonius circle, it is the locus of points,the ratio of whose distances from 2 fixed points is a constant(not equal to 1). This concept can be modified for a sphere. U can also solve the question by placing -3nC at (25,0,0)

Satvik Reddy - 7 years, 3 months ago

Place an arbitrary value of charge outside the sphere at an arbitrary position and find the potential on the surface of the sphere. Equate it to zero and you'll get the magnitude and position of the image charge such that the potential on the surface of the sphere is zero

Balaji Venkat - 7 years, 3 months ago

You can also watch some amazing videos on youtube.

Pinak Wadikar - 7 years, 1 month ago

Surround Ground

The answer is 18.

1 solution

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