Derive an expression for electric potential at a point due to a system of $\mathrm{N}$ charges.
Derive an expression for electric potential at a point due to a system of $\mathrm{N}$ charges.
Consider a system of charges $q_{1}, q_{2}, q_{3}, \ldots, q_{\mathrm{N}}$ with position vectors $r_{1}, r_{2}, r_{3}, \ldots, r_{\mathrm{N}}$
Electric potential at point $\mathrm{P}$ due to charge $q_{1}$,
$\mathrm{V}_{1}=\frac{k q_{1}}{r_{1 p}}$
where $k$ is coulomb constant $=\frac{1}{4 \pi \epsilon_{0}}$ and
$r_{1} \mathrm{P}=$ distance between charge $q_{1}$ and point $\mathrm{P}$.
Similarly electric potential due to charges $q_{2}, q_{3}, \ldots, q_{\mathrm{N}}$ are
$\mathrm{V}_{2}=\frac{k q_{2}}{r_{2 \mathrm{p}}}, \mathrm{V}_{3}=\frac{k q_{3}}{r_{3 \mathrm{p}}}$ and $\mathrm{V}_{\mathrm{N}}=\frac{k q_{\mathrm{N}}}{r_{\mathrm{NP}}}$
Electric potential is a scalar quantity. Hence total electric potential at $\mathrm{P}$ is, $\mathrm{V}=\mathrm{V}_{1}+\mathrm{V}_{2}+\mathrm{V}_{3}+\ldots ., \mathrm{V}_{\mathrm{N}}$
$\therefore \mathrm{V}=k\left[\frac{q_{1}}{r_{\mathrm{IP}}}+\frac{q_{2}}{r_{2 \mathrm{P}}}+\frac{q_{3}}{r_{3 \mathrm{P}}}+\frac{q_{\mathrm{N}}}{r_{\mathrm{NP}}}\right]$
$\therefore \mathrm{V}=k \sum_{i=1}^{\mathrm{N}} \frac{q_{i}}{r_{i \mathrm{P}}} \quad$ where $i=1,2,3, \ldots, \mathrm{N}$
Similar Questions
An arc of radius $r$ carries charge. The linear density of charge is $\lambda$ and the arc subtends a angle $\frac{\pi }{3}$ at the centre. What is electric potential at the centre
An arc of radius $r$ carries charge. The linear density of charge is $\lambda$ and the arc subtends a angle $\frac{\pi }{3}$ at the centre. What is electric potential at the centre
Write an equation for potential at a point in a uniformly charged spherical shell.
Write an equation for potential at a point in a uniformly charged spherical shell.
Find the potential $V$ of an electrostatic field $\vec E = a\left( {y\hat i + x\hat j} \right)$, where $a$ is a constant.
Find the potential $V$ of an electrostatic field $\vec E = a\left( {y\hat i + x\hat j} \right)$, where $a$ is a constant.
Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Given that $K=\frac{1}{4 \pi \varepsilon_0} \frac{q}{L^2}$, which of the following statement $(s)$ is (are) correct?
$(A)$ the elecric field at $O$ is $6 K$ along $O D$
$(B)$ The potential at $O$ is zero
$(C)$ The potential at all points on the line $PR$ is same
$(D)$ The potential at all points on the line $ST$ is same.
Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Given that $K=\frac{1}{4 \pi \varepsilon_0} \frac{q}{L^2}$, which of the following statement $(s)$ is (are) correct?
$(A)$ the elecric field at $O$ is $6 K$ along $O D$
$(B)$ The potential at $O$ is zero
$(C)$ The potential at all points on the line $PR$ is same
$(D)$ The potential at all points on the line $ST$ is same.
- [IIT 2012]
Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region: $Image$
$(A)$ the electrostatic field is zero
$(B)$ the electrostatic potential is constant
$(C)$ the electrostatic field is constant in magnitude
$(D)$ the electrostatic field has same direction
Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region: $Image$
$(A)$ the electrostatic field is zero
$(B)$ the electrostatic potential is constant
$(C)$ the electrostatic field is constant in magnitude
$(D)$ the electrostatic field has same direction
- [IIT 2013]