Superposition Principle and Continuous Charge Distribution

The superposition principle states that for a linear force system, the resultant force acting on a body is equal to the vector sum of all the individual stimulus. In Coulomb’s law the stimulus would be electrostatic force which depends upon the magnitude of charge and the distance between two charges.

The superposition principle is helpful when there are large number of charges in a system. Let’s consider the following case,

Principle of Superposition

For our convenience let us consider one positive charge, and two negative charges exerting a force on it, from the superposition theorem we know that the resultant force is the vector sum of all the forces acting on the body, therefore the force Fr , the resultant force can be given as follows,

\( \overrightarrow{F_r} ~=~ \frac {1}{4 \pi \in} \left[ \frac {Q q_1}{r^2_{12}} \hat{r}_{12} ~+~\frac {Qq_2}{r^2_{13}} \hat{r}_{13} \right] \)

Where,

\( \hat{r}_{12}\) and \( \hat{r}_{13}\) are the unit vectors along the direction of q1 and q2.

\( \in \) is the permittivity constant for the medium in which the charges are placed in.

Q, q1 and q2 are the magnitude of the charges respectively.

r12  and r13 are the distances between the charges Q and q1  &  Q and q2 respectively.

Continuous Charge Distribution:

We know that the smallest form of charge we can obtain would be +e or –e i.e. the charge of an electron or a proton, hence charges are quantized. Continuous charge distribution means that all charges are closely bound together having very less space between each other.

There are different ways in which charges can be distributed:

  1. Linear charge distribution.
  2. Surface charge distribution.
  3. Volume charge distribution.

Linear charge distribution:

Linear charge distribution is when the charges get distributed uniformly along a length, like around the circumference of a circle or along a straight wire, linear charge distribution is denoted by the symbol λ.

λ = dq/dl and it is measured in Coulombs per meter.

Surface charge distribution:

When a charge is distributed over a specific area, like the surface of a disk, it is called as surface charge distribution, it is denoted by Greek letter σ.

Surface charge distribution is measured Coulombs per square meter or Cm-2.

Volume charge distribution:

When a charge is distributed uniformly over a volume it is said to be volume charge distribution, like distribution of charge inside a sphere, or a cylinder. It is denoted by ρ.

Volume charge distribution is measured in coulombs per cubic meter or Cm-3.

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