2. Proof that the assembling energy for a sphere of radius \( R \) with uniformly distributed charge \( q \) is \[ U=\frac{3 q^{2}}{20 \pi \varepsilon_{0} R}(25 \%) \] Hint: Recall that the energy density of the electric field is \( u=\frac{\varepsilon_{0}}{2}|\vec{E}|^{2} \). 3. If the sphere rotates with ancular
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Since the charge is uniformly distributed, the electric field at a distance \( r \) from the center of the sphere is given by \( E = \frac{1}{4\pi\epsilon_0} \frac{q}{R^3} r \) for \( r \leq R \). Show more…
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In this problem you will calculate the electrostatic energy of a uniform sphere of charge $Q$ with radius $R .$ The electric potential $V(r)$ at any radius $r<R$ is given by (16.49) in Problem $16.28$, and the potential energy of a charge $q$ at radius $r$ is $q V(r)$. (a) Write down the total charge contained between radius $r$ and $r+d r$, and hence find the potential energy of that charge. (b) If you integrate your answer to (a) from $r=0$ to $r=R$, you will get twice the total potential energy of the whole sphere since you will have counted the energy of any two charge elements twice. Show that the total Coulomb energy of the sphere is $$ U_{\text {Coul }}=\frac{3}{5} \frac{k Q^{2}}{R} $$
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(1I) Show that the electrostatic energy stored in the electric field outside an isolated spherical conductor of radius $r_{0}$ carrying a net charge $Q$ is $$U=\frac{1}{8 \pi \epsilon_{0}} \frac{Q^{2}}{r_{0}}.$$ Do this in three ways: (a) Use Eq. 6 for the energy density in an electric field [Hint: Consider spherical shells of thickness $d r ] ;(b)$ use Eq. 5 together with the capacitance of an isolated sphere (Section 2 of "Capacitance, Dielectrics, Electric Energy Storage");(c) by calculating the work needed to bring all the charge $Q$ up from infinity in infinitesimal bits $d q$ .
A sphere of radius $R$ carries total charge $Q$ distributed uniformly over its surface. Show that the energy stored in its electric field is $U=k Q^{2} / 2 R$
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