Question

Derive an expression for the electric potential in a electric field of positive point charge at distance r.​

Answer 1

$\large \dag$ Defination :-

Electric potential at a point may be defined as Work done in bringing a unit positive test charge from infinity to that point without changing kinetic energy.

⇝ In Attached Figure :-

• q = Point charge due to which electric potential is to be find.

• $\large\sf+ q_ \circ$ = +ve Test charge

• P = Any point at distance r where potential is to be find.

$\large \dag$ Derivation :-

[ Figure in Attachment ]

Small work done to move charge from point P to infinity is :

$\text{dW = - F.dx}$

$:\longmapsto \rm{dW = - q_{\circ}E .dx}$

$:\longmapsto\text{dW} = - \frac{\text{kqq}_{\circ}}{\text x {}^{2} }$

⏩ Integrating Both Side ;

$\text{W}_{\text P\rightarrow \infty } = - \int\limits^ \infty _\text r \text{kqq}_{\circ}( {\text x}^{ - 2} ) \ \text {dx}$

$= - \text{kqq}_{\circ} {{\bigg[ - \frac{1}{\text x} \bigg]}_\text r^ \infty }$

$\text{W}_{\text P \rightarrow \infty } = - \frac{\text{kqq}_{\circ}}{\text r}$

$\large\red{\therefore \: \boxed{ \boxed{\text{W}_{\infty \rightarrow\text P } = \frac{\text{kqq}_{\circ}}{\text r} }}}$

⏩ By Defination :

$:\longmapsto \rm V_P=\frac{ W_{\infty \rightarrow P} }{q_{\circ}}$

$:\longmapsto \rm V_P = \frac{{kqq}_{\circ}}{rq_{\circ}}$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf V_P = \frac{{kq}}{r}} }}}$

which is electric potential due to a Positive point charge at point P which is r distance away from Positive Charge.

Answer 2

Answer:

$\sf\tt\large{\green {\underline {\underline{⚘\;Question:}}}}$

• Derive an expression for the electric potential in a electric field of positive point charges at distance r.

$\sf\tt\large{\green {\underline {\underline{⚘\;Answer:}}}}$

$\sf\tt\large{\red {\underline {\underline{⚘\;The \;expression \;for \;electric \; potential \;in \;a \;electric field \;positive \;point \;charge \; at \;distance \; r is:}}}}$

• $Vp = \frac{kq}{r}$

$\sf\tt\large{\blue {\underline {\underline{⚘\;Solution:}}}}$

• Refer the given above attachment for more information and for better understanding.

Hope it helps u mate.

Thank you .