Electric Field due to Point Charge and System of Charges

Electric Field due to Point Charge and System of Charges

In this article, we will rigorously derive expressions for the Electric Field due to Point Charge and System of Charges. We will review the Coulomb’s quantitative statement about the force between two point charges. However, we have to understand the definition of point charge before we start our discussion. When the dimension of the charged particles is much smaller than the separation between them we can consider them as point charges. Thus point charge may not be an absolutely zero-dimensional object.

Coulomb’s Law

The most fundamental law of electrostatics was first published in 1785 by French physicist Charles-Augustin de Coulomb and was essential to the development of the theory of electromagnetism. This is an inverse-square law and is analogous to Isaac Newton’s inverse-square law of universal gravitation. According to Coulomb’s law

The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.
The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive. If they have different signs, the force between them is attractive.

Thus if two point charges q_1 and q_2 are separated by a distance r in the vacuum then the magnitude of the force between them is given by

    \[  F= k \frac{q_1q_2}{r^2} \]

here k is a proportionality constant and is usually put as k=\frac{1}{4\pi\epsilon_0}. \epsilon_0 is called the permittivity of free space. The value of \epsilon_0 in SI unit is 8.854\times10^{-12}C^2N^{-1}m^{-2}. Thus in vector notation, Coulomb’s law can be expressed as

    \[  \boxed{\vec{F}= \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2}\hat{r}} \]

Electric Field due to a Point Charge

An electric field is a vector field that associates the Coulomb force experienced by a test charge at each point in space to the source charge. The strength and the direction of the electric field can be determined from the Coulomb force F on a test charge q. If the field is generated by a positive source point charge Q, the direction of the electric field points along lines directed radially outwards from it. Similarly, for a negative point source charge, the direction is radially inwards. The magnitude of the electric field E can be derived from Coulomb’s law. Let there is a point charge Q placed in the vacuum. We introduce another point charge q (or test charge) at a distance r from the charge Q. The electric field at point P due to the point charge Q is given by

    \[\vec{E}=\frac{\vec{F}}{q}\]

direction of electric field due to point charge

In the figure above we have shown the direction of the electric field due to a positive point charge Q. The direction of \vec{r} is also shown. The magnitude of the electric field is proportional to the length of the \vec{E} shown. However, if a relatively large test charge q is brought within the vicinity of the source charge Q, it is bound to modify the original electric field due to the source charge. A simple way to avoid this conflict is to use a negligibly small test charge q. Thus our definition of electric field modifies to,

    \[\vec{E}=\lim_{q\to 0} \left(\frac{\vec{F}}{q}\right)\]

According to the definition given above, the electric field at point P due to the point charge Q is

    \[  \boxed{\vec{E}= \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}\hat{r}} \]

S.I. unit of electric field is N C^{-1}.

In the diagram below, we have shown the electric field due to a positive point charge as a function of distance.Electric field due to point charge as a function of distance

Salient Features of Electric field due to Point Charge

  • An electric field due to a point charge is a vector quantity.
  • This electric field is the property of the source charge and does not depend on the test charge.
  • For a positive source charge, the electric field will point radially outward from the source charge and for a negative one, it will direct radially inwards.
  • The magnitude of the electric field due to a point charge also depends on the distance from the source charge.
  • Electric field due to point charge has spherical symmetry and does not depend on \hat{\theta}, \hat{\phi}.
  • Electric field at any point in space is tangent to the line of force at that point.

Electric Field due to a System of Charges

Let us consider a system of n number of charges denoted as Q_1,q_2,Q_3,...Q_n with position vectors \vec{r_1},\vec{r_2},\vec{r_3},... \vec{r_n} respectively. We have to calculate the electric field at a point P(\vec{r}) due to this discrete charge distribution. We introduce a test point charge q at a position P(\vec{r}). The individual electric force at the test point P can be calculated according to the Columb’s law shon in the section above. The total electric force at the test point P can be calculated using the superposition principle. According to which, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. Electric force \vec{F_1} at point P due to charge Q_1 at position \vec{r_1} is

    \[\vec{F_1}=\frac{Q_1q}{4\pi\epsilon_0 r_{1P}^2}\hat{r}_{1P}\]

We have shown the direction of the force vector in the figure below. Similarly, the electric force \vec{F_2} at the same point due to charge Q_2 is given by

    \[\vec{F_1}=\frac{Q_2q}{4\pi\epsilon_0 r_{2P}^2}\hat{r}_{2P}\]

Thus using the superposition principle we can calculate the total electric force at point P due to the system of charges.

    \[\vec{F}(\vec{r})=\vec{F_1}+\vec{F_2}+...+\vec{F_n}\]

and by definition of an electric field, the total electric field at point P due to this system of charges becomes

    \[\vec{E}(\vec{r})=\lim_{q\to 0} \left(\frac{\vec{F}}{q}\right)\]

    \[=\frac{1}{4\pi\epsilon_0}\left[\frac{Q_1}{r_{1P}^2}\hat{r}_1P+\frac{Q_1}{r_{2P}^2}\hat{r}_2P+...+\frac{Q_1}{r_{nP}^2}\hat{r}_nP\right]\]

    \[\boxed{\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}\sum_{i=1}^{n}\frac{Q_i}{r_{iP}^2}\hat{r}_{iP}}\]

electric field due to two point charges

In the diagram above, we have shown the electric field due to a system of two source point charges Q_1, and Q_2. The resuslant electric field \vec{E} is the vector sum of \vec{E_1} and E_2. We also have shown the direction of \vec{r_1} and \vec{r_2}.

Help us to improve. Rate this article. It matters.
1 Star2 Stars3 Stars4 Stars5 Stars (3 votes, average: 4.67 out of 5)

Loading...