5. A spherically symmetric charge distribution has the following volume charge density p: er/R r < R r > R p(r) = where y is a constant. a) What are the units of the constant ? b) What is the total charge Q contained in the distribution? c) Using the integral form of Gauss's law find the electric field in the regions r < R and r > R. For r > R, express your answer in terms of the total charge Q calculated in part b). d) Now use the local form of Gauss's law along with spherical symmetry to find the electric field in the region r < R. From the local form of Gauss's law you should find an expression that can be integrated (radially). You can integrate this expression from 0 to r to find Ē, but you might find there is some ambiguity. Instead, another approach is to do an indefinite integral and require the electric field to be continuous at r = R (for the r > R region you can just use your answer from part c)). In general, using the local form is more challenging technically so I'm sweeping some mathematical points under the rug.
5. A spherically symmetric charge distribution has the following volume charge density p: er/R r < R r > R p(r) = where y is a constant. a) What are the units of the constant ? b) What is the total charge Q contained in the distribution? c) Using the integral form of Gauss's law find the electric field in the regions r < R and r > R. For r > R, express your answer in terms of the total charge Q calculated in part b). d) Now use the local form of Gauss's law along with spherical symmetry to find the electric field in the region r < R. From the local form of Gauss's law you should find an expression that can be integrated (radially). You can integrate this expression from 0 to r to find Ē, but you might find there is some ambiguity. Instead, another approach is to do an indefinite integral and require the electric field to be continuous at r = R (for the r > R region you can just use your answer from part c)). In general, using the local form is more challenging technically so I'm sweeping some mathematical points under the rug.
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part d please
Transcribed Image Text:5. A spherically symmetric charge distribution has the following volume charge density p:
Le-T/R r < R
p(r) = To
r> R
where is a constant.
a) What are the units of the constant ?
b) What is the total charge Q contained in the distribution?
c) Using the integral form of Gauss's law find the electric field in the regions r < R and r > R. For r > R, express your
answer in terms of the total charge Q calculated in part b).
d) Now use the local form of Gauss's law along with spherical symmetry to find the electric field in the region r < R.
From the local form of Gauss's law you should find an expression that can be integrated (radially). You can integrate
this expression from 0 to r to find Ē, but you might find there is some ambiguity. Instead, another approach is to do an
indefinite integral and require the electric field to be continuous at r = R (for the r > R region you can just use your
answer from part c)). In general, using the local form is more challenging technically so I'm sweeping some mathematical
points under the rug.
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