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Potential Energy In An External Field

Potential Energy In An External Field :- An external field is a force field that exists independently of the object experiencing it. It can be :

Gravitational field, created by massive bodies like the Earth or the Sun.
Electric field, generated by charges or varying magnetic fields.
Magnetic field, produced by moving charges or magnetic materials.

These fields exert forces on objects within their influence, which can change the object’s energy without direct contact.

(a) Potential Energy of a Single Charge in an External Field

(Potential Energy In An External Field)

We have already discussed the potential energy of a system of charges, where the source of the electric field was known and consisted of specified charges at known locations.

Now, we consider the case where a single charge (or multiple charges) is placed in an external electric field E, which is not produced by the charges whose potential energy we want to calculate. The sources creating this external field are not of interest—they are unknown or irrelevant to the current analysis. The external electric field E and the corresponding electric potential may vary from point to point in space.

By definition, the work done in bringing a unit positive charge from infinity to a point P in the field is equal to the electric potential at point P. Thus, the work done in bringing a charge from infinity to point in the external field is :

$\displaystyle W=q.V(\overrightarrow{r})$

This work is stored as the potential energy of the charge at that location. Therefore :

$\displaystyle U=q.V(\overrightarrow{r})$

(b) Potential Energy of a System of Two Charges in an External Field

(Potential Energy In An External Field)

Let and be two point charges placed at position vectors $r_{1}$ and $r_{2}$ respectively, in an external electric field of intensity $E$ . We want to compute the total potential energy of the system.

(1) Work done in bringing charge from infinity to position $r_{1}$ :

$\displaystyle {{W}_{1}}={{q}_{1}}.V(\overrightarrow{{{r}_{1}}})$

(2) Work done in bringing charge from infinity to position $r_{2}$ includes two contributions :

(i) Work done against the external electric field E :

$\displaystyle {{W}_{2}}={{q}_{2}}.V(\overrightarrow{{{r}_{2}}})$

(ii) Work done against the electric field produced by charge :

$\displaystyle {{W}_{3}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{1}}{{q}_{2}}}{{{r}_{12}}}$

where $\displaystyle {{r}_{12}}=\left| \overrightarrow{{{r}_{2}}}-\overrightarrow{{{r}_{1}}} \right|$ is the distance between the two charges.

By the superposition principle, the total potential energy (U) of the system is the sum of all the work done :

$\displaystyle U={{W}_{1}}+{{W}_{2}}+{{W}_{2}}={{q}_{1}}.V(\overrightarrow{{{r}_{1}}})+{{q}_{2}}.V(\overrightarrow{{{r}_{2}}})+\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{1}}{{q}_{2}}}{{{r}_{12}}}$

Potential Energy In An External Field

(a) Potential Energy of a Single Charge in an External Field

(Potential Energy In An External Field)

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