Gauss's Law and Superposition of Fields

AI Thread Summary
The discussion focuses on applying Gauss's Law to calculate the electric fields generated by a charged shape and a negatively charged sphere. The charge density is derived as 6Q/(7πR^3), leading to a total charge of 8/7Q for the filled sphere. The electric field at the surface of this sphere is calculated to be 2/(7πε₀R²). For the negatively charged sphere, the charge is determined to be -Q/7, resulting in an electric field of -Q/(7πε₀R²). The final electric field at point P is the sum of both fields, with a correction noted regarding a factor of 4 in the answer.
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Gauss's Law and Superposition of Fields (edited again, something else wrong)

Homework Statement


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Right. The shape itself has charge Q, so it has charge density \frac{Q}{\frac{4}{3} \pi R^3 - \frac{4}{3} \pi (\frac{R}{2})^3} = \frac{6Q}{7\pi R^3} Let's call this \rho. If it were filled in entirely, then, it would have charge:

Q + \rho V = Q + \frac{6Q}{7\pi R^3}\cdot \frac{4}{3}\pi\left(\frac{R}{2}\right)^3 = \frac{8}{7} Q. (V being the volume of the small sphere.)

By Gauss's Law, the field at the surface of this filled sphere would be:

E = \frac{\frac{8}{7}Q}{4 \pi \epsilon_0 R^2} = \frac{2}{7 \pi \epsilon_0 R^2}

Considering now the smaller, negatively charged sphere, this would be carrying a charge of

-\rho V = -\frac{6Q}{7\pi R^3} \cdot \frac{4}{3}\pi \left(\frac{R}{2}\right)^3 = -\frac{Q}{7}.

The electric field at its surface would then be \frac{\frac{-Q}{7}}{4 \pi \epsilon (\frac{R}{2})^2} = -\frac{Q}{7 \pi \epsilon _0 R^2}.

The electric field at P of the shape, then, is the sum of the individual fields of the filled sphere and the negatively charged sphere - I don't get the answer I'm meant to, and I can't see what I've done wrong. It's probably something really stupid.
 

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You have the correct answer. Note that (2/7) + (-1/7) = 1/7, which is the answer you are meant to get.


Oh, by the way, note that the answer you are meant to get has a 4 in the numerator and in the denominator, so multiply and divide your answer by 4. :wink:
 
Oh yeah. I didn't notice that 4 in the denominator. Knew it was something stupid! Cheers.
 
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