The electric potential due to a certain configuration is given by-
V(r)=Ae to the power of negative k r end exponent/r.
where A and K are constants.
Find the volume charge density p(r).

Asked by Aryan Bhatia | 13th May, 2015, 10:16: PM

Expert Answer:

begin mathsize 14px style Given space that space the space electric space potential comma space straight V left parenthesis straight r with rightwards arrow on top right parenthesis space equals fraction numerator straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r end fraction We space know space that space electric space field comma space straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis space equals negative triangle straight V left parenthesis straight r with rightwards arrow on top right parenthesis straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis space equals negative fraction numerator partial differential open square brackets fraction numerator straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r end fraction close square brackets over denominator partial differential straight r end fraction space straight r with hat on top straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis space equals negative fraction numerator partial differential open parentheses straight A space straight e to the power of negative straight k space straight r end exponent. straight r to the power of negative 1 end exponent close parentheses over denominator partial differential straight r end fraction space straight r with hat on top straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis space equals negative space open square brackets fraction numerator negative straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r squared end fraction minus fraction numerator straight k space straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r end fraction close square brackets space straight r with hat on top straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis space equals open square brackets fraction numerator straight k space straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r end fraction plus fraction numerator straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r squared end fraction close square brackets space straight r with hat on top We space know space that space increment. straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis space space equals fraction numerator straight rho left parenthesis straight r with hat on top right parenthesis over denominator straight epsilon subscript 0 end fraction Charge space density space straight rho left parenthesis straight r with hat on top right parenthesis equals straight epsilon subscript 0 space open square brackets increment. straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis close square brackets Substituting space for space straight E with rightwards arrow on top left parenthesis straight r with rightwards arrow on top right parenthesis space we space get space that colon straight rho left parenthesis straight r with hat on top right parenthesis equals straight epsilon subscript 0 space open square brackets increment. open square brackets fraction numerator straight k space straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r end fraction plus fraction numerator straight A space straight e to the power of negative straight k space straight r end exponent over denominator straight r squared end fraction close square brackets space straight r with hat on top close square brackets straight rho left parenthesis straight r with hat on top right parenthesis equals straight epsilon space subscript 0 open square brackets increment. open parentheses 1 plus kr close parentheses space straight e to the power of negative straight k space straight r end exponent fraction numerator straight r with hat on top over denominator straight r squared end fraction close square brackets straight rho left parenthesis straight r with hat on top right parenthesis equals straight epsilon subscript 0 straight A space open square brackets open parentheses 1 plus kr close parentheses space straight e to the power of negative straight k space straight r end exponent open parentheses increment. fraction numerator straight r with hat on top over denominator straight r squared end fraction close parentheses plus fraction numerator straight r with hat on top over denominator straight r squared end fraction. increment open parentheses 1 plus kr close parentheses space straight e to the power of negative straight k space straight r end exponent close square brackets straight rho left parenthesis straight r with hat on top right parenthesis equals straight epsilon subscript 0 straight A open square brackets 4 πδ cubed left parenthesis straight r with rightwards arrow on top right parenthesis open parentheses 1 plus kr close parentheses space straight e to the power of negative straight k space straight r end exponent minus fraction numerator straight k squared straight e to the power of negative straight k space straight r end exponent over denominator straight r end fraction close square brackets end style
Note:Solution to this query contains concepts and divergence formulaes which you would come across in higher standards.

Answered by Jyothi Nair | 14th May, 2015, 12:13: PM