Question: Prove that E is a conservative force field with the potential function u: u ( x;y;z ) = q ; 4 0 jx x 0

Prove that E is a conservative force field with the potential function u: u

(

x;y;z

) =

q ;

40

0

j in the sense E

=

.

In this sense, the functionU

(

x;y;z

) =

(

x;y;z

) =

4 " 0

0

jis known as the potential energy of a charge located at

(

x;y;z

) .

)

Supposetwoelectricchargesqand

qarelocatedatthepoints

(1

;

0

;

0))

and

(1

;

0

;

0),

respectively. Determine the electric field E and the associated potential energy U at an arbitrary point

(

x; y; z

)

different from

(1

;

0

;

0)

(1

;

0

;

0) .

Hint: The electric field and potential energy obey the principle of superposition, meaning they result from the sum of the fields and potentials of each charge.

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