(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A static charge distribution has a radial electric field of magnitude

##E = \alpha \frac{e^{-\lambda r}}{r} ##

where λ and α are positive constants. Calculate the total charge of the distribution.

2. Relevant equations

Gauss's law ##Q/\epsilon_0 = \int \vec{E} \cdot d\vec{S}##

##\rho/\epsilon_0 = \nabla \cdot \vec{E} ##

3. The attempt at a solution

I have tried two ways to go about this problem, without success. First I tried using Gauss's law, placing a Gaussian sphere of radius a on the distribution:

##Q/\epsilon_0 = \int \vec{E} \cdot d\vec{S} = 4\pi\alpha (ae^{-\lambda a}) ##

We should now find the total charge by letting ## a \rightarrow \infty ##, but ##\lim_{a \rightarrow \infty} ae^{-\lambda a} = 0##, giving ##Q = 0## which i thought cannot be right.*

My second method involved finding the charge distribution

##\rho/\epsilon_0 = \nabla \cdot \vec{E} = \alpha \frac{e^{-\lambda r}(1-\lambda r)}{r^2} ## but this gives an integral which is divergent!

*I have done some further thinking and realised that ##Q_{tot} = 0## might be the answer, because the charge density changes sign at ##r = 1/\lambda## so the total net charge could as well be zero.

However, how is it possible that the volume integral of #\rho# is divergent then? Or is it not?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Finding Total Charge from E-field

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**