Half Life of Charge on metal ball.

Electricity and Magnetism Level 3

Let a metal ball of Radius r r is placed in infinite homogeneous conducting medium having dielectric constant k k , resistivity ρ \rho . Let at time t=0 , Q charge is given to ball , then find the half Life of the charge present on sphere.

Inspired from Raghav's Note
It is simple version of that problem.

Yes Answer is independent of Size of ball...

k ρ ϵ ln ⁡ 3 3 \cfrac { k\rho \epsilon \ln { 3 } }{ 3 } k ρ ϵ ln ⁡ ( 2 + 2 ) k\rho \epsilon \ln { (2+ } \sqrt { 2 } ) 2 k ρ ϵ ln ⁡ 2 2k\rho \epsilon \ln { 2 } k ρ ϵ ln ⁡ 3 k\rho \epsilon \ln { 3 } k ρ ϵ ln ⁡ 2 2 \cfrac { k\rho \epsilon \ln { 2 } }{ 2 } k ρ ϵ ln ⁡ 2 k\rho \epsilon \ln { 2 }

Karan Shekhawat by Karan Shekhawat , 25, India

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1 solution

Karan Shekhawat
Apr 21, 2015

I did in this way....

R e q = ∫ r ∞ ρ d x 4 π x 2 R e q = ρ 4 π r \displaystyle{{ R }_{ eq }=\int _{ r }^{ \infty }{ \cfrac { \rho dx }{ 4\pi { x }^{ 2 } } } \\ \boxed { { R }_{ eq }=\cfrac { \rho }{ 4\pi r } } }

Now draw it's Eqv. cct. diagram.

Let at t=t , charge on ball is Q − q Q-q , then current is I = − d q d t I=\cfrac { -dq }{ dt } . Now use ohm's law ::

\displaystyle{I=\cfrac { -dq }{ dt } \\ \Delta V=I\times { R }_{ eq }\\ \left\{ \because \Delta V={ V }_{ r }-{ V }_{ \infty }=\cfrac { Q-q }{ 4\pi \epsilon kr } \right \\ \\ -\left\{ \cfrac { \rho }{ 4\pi r } \right\} \cfrac { dq }{ dt } =\cfrac { Q-q }{ 4\pi \epsilon kr } \\ \Rightarrow \boxed { t=k\epsilon \rho \ln { 2 } } }

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