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Self Energy | Self Energy Of A Charged Body

Self Energy | Self Energy Of A Charged Body :- The self energy of a charged body refers to the amount of work required to charge the body by bringing in infinitesimal elements of charge from infinity and distributing them to form the complete charged object, against their mutual electrostatic repulsion.

Actually the work done by the external agent to charge a body, results in an electric field around that body. This whole work is to create this electric field. So in simple terms self energy is the electrostatic potential energy stored in the electric field around a charged body.

(1) Self Energy Of A Conducting Sphere (solid or hollow) & Non-Conducting Hollow Sphere

We have discussed that whenever a group of electric charges are brought together to form a system, work must be done in the process. This work gets stored as the system’s electrostatic potential energy. Let’s now look at a specific case : charging a spherical shell with radius

Self Energy | Self Energy Of A Charged Body - Curio Physics

While charging the sphere, we gradually bring small amounts of charge to it from infinity. The charge already present on the sphere resists the addition of each new element of charge. Suppose at a certain moment the sphere has a charge ; at that point, the electric potential of the sphere is given by :

$\displaystyle V=\frac{kq}{R}$

If now a charge dq is brought to its surface from infinity then small amount of work done in this process :

$\displaystyle dW=Vdq=\frac{kq}{R}dq$

Total work done in charging the sphere with a total charge Q can be given as

$\displaystyle W=\int{dW=\int\limits_{0}^{Q}{\frac{kq}{R}dq=}}\frac{k}{R}\int\limits_{0}^{Q}{qdq}$

$\displaystyle \therefore W=\frac{k{{Q}^{2}}}{2R}$

Above equation is the total work to done in charging the sphere and hence it is the required expression for self energy of a conducting sphere (solid or hollow) & non-conducting hollow sphere.

$\displaystyle {{U}_{self}}=\frac{k{{Q}^{2}}}{2R}$ …..(1)

Now let us prove that self energy is also equal to the energy stored in the electric field around a charged body.

Now the sphere is having a total charge Q on its surface and an electric field exist in its surrounding from its surface to infinity(since inside there is no electric field). Let us calculate the field energy associated with this charged sphere.

Consider an elemental spherical shell of radius x and thickness dx as shown in the figure below :

Electric field intensity due to the charged sphere at a distance x from its centre O is

$\displaystyle E=\frac{kQ}{{{x}^{2}}}$

We know that energy density of electric field (u) is given by :

$\displaystyle u=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}$

Volume (dV) of the elemental spherical shell thickness dx :

$\displaystyle dV=4\pi {{x}^{2}}dx$

Field energy stored in the elemental shell :

$\displaystyle dU=udV=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}\times 4\pi {{x}^{2}}dx=\frac{1}{2}{{\varepsilon }_{0}}{{\left( \frac{kQ}{{{x}^{2}}} \right)}^{2}}\times 4\pi {{x}^{2}}dx$

$\displaystyle \Rightarrow dU=\frac{k{{Q}^{2}}}{2{{x}^{2}}}dx$

Thus the total field energy stored in the electric field can be calculated by integrating above expression of dU from the surface of the sphere to infinity as electric field inside the sphere is zero.

$\displaystyle {{U}_{self}}=\int{dU}=\int\limits_{R}^{\infty }{\frac{k{{Q}^{2}}}{2{{x}^{2}}}dx=\frac{k{{Q}^{2}}}{2}\int\limits_{R}^{\infty }{{{x}^{-2}}}}dx=\frac{k{{Q}^{2}}}{2}\left[ -\frac{1}{x} \right]_{R}^{\infty }$

$\displaystyle \therefore {{U}_{self}}=\frac{k{{Q}^{2}}}{2R}$ …..(2)

This result matches equation (1). It shows that the total work done in charging a body is stored in its electric field as field energy, known as the body’s self-energy. Once charged, if the body’s shape and charge distribution remain unchanged, its self-energy stays constant. So, the self-energy of a charged body is the total energy stored in the electric field it creates.

(2) Self Energy Of A Uniformly Charged Non-Conducting Solid Sphere

Let us consider a uniformly charged non-conducting solid sphere of radius R having total charge Q. Now outside the non-conducting sphere, the electric field intensity is same as in case of a conducting sphere of same radius. Thus field energy in the surrounding of non-conducting sphere from its surface to infinity is same as that of conducting sphere.

$\displaystyle {{\left[ {{U}_{self}} \right]}_{R\to \infty }}=\frac{k{{Q}^{2}}}{2R}$ …..(3)

Now unlike a conducting sphere, in the interior of a non-conducting uniformly charged sphere, the electric field is non-zero. Thus the field energy also exist in the interior region. Inside a non-conducting uniformly charged insulating sphere the electric field intensity at a distance x from the centre is given by :

$\displaystyle {{E}_{inside}}=\frac{kQ}{{{R}^{3}}}x$

Consider an elemental shell of thickness dx and radius x in the interior of insulating sphere as shown in figure below :

Field energy stored in the elemental shell :

$\displaystyle dU=udV=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}\times 4\pi {{x}^{2}}dx=\frac{1}{2}{{\varepsilon }_{0}}{{\left( \frac{kQ}{{{R}^{3}}}x \right)}^{2}}\times 4\pi {{x}^{2}}dx$

$\displaystyle \Rightarrow dU=\frac{k{{Q}^{2}}}{2{{R}^{6}}}{{x}^{4}}dx$

Total field energy inside the sphere :

$\displaystyle {{\left[ {{U}_{self}} \right]}_{0\to R}}=\int{dU}=\int\limits_{0}^{R}{\frac{k{{Q}^{2}}}{2{{R}^{6}}}{{x}^{4}}dx=\frac{k{{Q}^{2}}}{2{{R}^{6}}}\int\limits_{0}^{R}{{{x}^{4}}}}dx=\frac{k{{Q}^{2}}}{2{{R}^{6}}}\left[ \frac{{{x}^{5}}}{5} \right]_{0}^{R}$

$\displaystyle \Rightarrow {{\left[ {{U}_{self}} \right]}_{0\to R}}=\frac{k{{Q}^{2}}}{10R}$ …..(4)

Thus total self energy of insulating sphere :

$\displaystyle {{U}_{self}}={{\left[ {{U}_{self}} \right]}_{0\to R}}+{{\left[ {{U}_{self}} \right]}_{R\to \infty }}=\frac{k{{Q}^{2}}}{10R}+\frac{k{{Q}^{2}}}{2R}$