Electromagnetic Theory 1

Transcription

1 / lectomagnetic Theoy uestion : lectostatic Potential negy A sphee of adius caies a positive chage density ρ constant Obviously the spheical coodinates system is appopiate hee Take - C m - and cm τ a) Compute the total chage ρ d whee sin θ d dθ, whee is a b) how that the magnitude of the electic field of this distibution is given by: c) Compute the electic potential in the egions and d) Calculate (5 cm) e) se the definition of the enegy of the configuation ode to show that: Hint: integals f) Calculate olution: 6 6 all spcae, ie you have to calculate the two a) - Computing the total chage of the sphee ρ d sin θ dθ sin θ dθ d ρ () C m ( m) C nc a) - Computing the lectic field in the egion ( the sphee) We simply use Gauss s law to do so An appopiate gaussian suface is a sphee of adius ( ) (ee Fig A) pheical coodinates ae the appopiate system hee We have:, whee ( ) Gauss s law () in Physics Dept Yamouk nivesity - Phys, Fist / Fist xam at // /5

2 Fom the figue we can easily see that: d // which means that ( ) is adial, ie a, and because of the spheical symmety magnitude of the vecto Thus we can wite the lhs of q as: (the ) is constant fo all field points, (see P, P and P ) Fig A ( ) Fig B () Now we compute (hs of q ) which denotes the chage losed in the gaussian suface θ ρ ( ) sin θdθ φ θ φ quating q and q we get: d d sin θdθ () ( ) ( ) (5) and as we said ealie, the electic filed is adial, thus we have: ( ) - Computing the lectic field in the egion ( the sphee) Again an appopiate gaussian suface is a sphee of adius ( ) (ee Fig B) We have:, whee ( ) Gauss s law () sing the same aguments as fo the egion, we can wite the lhs as: (6) Physics Dept Yamouk nivesity - Phys, Fist / Fist xam at // /5

3 Physics Dept Yamouk nivesity - Phys, Fist / Fist xam at // /5 ouside (8) in the hs of q, which denotes the chage losed in the gaussian suface is, nothing else but, the total chage of the sphee, ie Thee is no chage in the egion Finally we have () The electic filed being adial, we have () q shows, as expected, that the electic field a spheical distibution of chages depends on / c) - Computing the electic potential in the egion ( the sphee) d d out () The esult in q shows, as expected, the depende on / of the electic potential a spheical distibution of chage In paticula we have d d () Note that q can be obtained fom q by eplacing by c) - Computing the electic potential in the egion ( the sphee) d d d d d () In ode to obtain in the egion ( ), we need to calculate two integals We have calculated the fist one (q ) Fo the second integal we have:

4 Physics Dept Yamouk nivesity - Phys, Fist / Fist xam at // /5 d d d d () Adding q and q we get the potential the sphee: (5) Note that, as expected, is positive and the chaged sphee xecise: Check the dimensionality of q 5 d) Computing (5 cm) (We ae the sphee!) cm e) Computing the enegy of the configuation τ space all d d d (6) d d () d d (8) The enegy of the configuation is thus: 8 8 τ d space all () f) Numeical application: fo - C m -, cm, we have: m J J () Note: is also given by:

5 I ρ 6 6 ( ) ρ ( < < ) ρ I I ( ) ρ ( ) 6 ( ) d 6 6 I ( ) ρ τ d, () because ρ( ) ; No chage the sphee! () d () () Physics Dept Yamouk nivesity - Phys, Fist / Fist xam at // 5/5