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  Potential for Continuous Charge Distribution and Energy, Physics tutorial

Potential due to an Infinite Line Charge:

We are familiar with the expression for the electric field at a point near an infinitely long charged wire (or a line charge) as an application of gauss's law.

We state that:

E = (λ/2πεor) rˆ

Here 'λ' is the charge per unit length on the wire, 'r' is the perpendicular distance of the point from the wire, 'εo' is the permittivity of free space, and r ˆ is a unit vector all along the direction of increasing 'r'.

We wish for to drive an expression for the potential due to the wire at point 'P'.

We know that the negative of line integral of the electric field between infinity and any point provides the value of the potential at that point, that is,

Φr = - r∫∞ E. dr

We know compute the line integral by first taking a finite distance 'r', rather than infinity and then letting r1 go to infinity. Here r1 is the distant of the point Q from the wire. The integral then provides us the difference in potentials between p and Q, that is:

Φr - Φr1 = - r1∫r E .dr

Replacing the expression for 'E' from the equation above, that is:

E = (λ/2πεor) rˆ

Φr - Φr1 = - (λ/2πεo) r1∫r (Y dr/r)

Since Y and dr are in similar direction, we have

Φr - Φr1 = - (λ/2πεo) r1∫r (dr/r)

Φr - Φr1 = - (λ/2πεo) ln (r/r1) = (λ/2πεo) ln (r/r1)

Now let us attempt to compute the potential with respect to infinity by letting r1 go to infinity. We notice from the equation E = (λ/2πεor) rˆ that Φr1 anywhere in the vicinity of the linear charge distribution (r finite), goes to infinity. This is due to the reason that the supposition of a uniform and infinite charge per unit length over an infinitely long line invariably leads to the infinite amount of charge. Thus, the sum of finite contributions from each portion of an infinite amount of charge leads to the infinite potential.

Equipotential surface of a Uniformly Line Charge:

By now we have comprehend that an equinoctial surface is the locus of all point having similar potential. For a uniform infinite line charge, the potential at a distance r is given by the equation:

Φr - Φr1 = (λ/2πεo) ln (r/r1)

From the above we can observe that the electric potential is similar for all points that are equidistant from the line of charge. Thus, the Equipotential are cylindrical by the line of charges as the axis of cylinder

Potential of a Charged Circular disc:

1338_Potential of a Charged Circular disc.jpg

The electric potential due to a continuous charge distribution can in principle be found through integration by employing the potential of a point charge. Though, this is helpful merely in simple configurations where the integration can be carried out. An illustration is a circular disk of radius 'R' carrying a uniform surface charge density 'σ' and we wish to determine the potential at a point on the axis at a distance 'z' from the center.

Split the disk into thin rings of radius 'R' and thickness dR'. The charge on the thin ring is:

dq = σ (2πRdR')

As the point P is at distance √(R2 + z2) from each and every point on the thin ring, the potential there due to the thin ring is:

dV = (kdq)/(√ R2 + z2) = (2πσkrdR')/(√ R2 + z2)

The potential at 'P' due to the whole disk is:

V = 2πσk 0∫k (rdR')/(√ R2 + z2)

By employing the substitution u = r2, du = 2rdR', we determine:

V = πσk 0∫R2 (du)/(√u2 + z2) = 2πσk [√(a2 + z2) -√z2]

Note that we take the positive sign for the square roots.

If the point 'P' is far away, we can take the limit z → ∞

V = 2πσk [√z2 {1 + (R2/z2)1/2} - √z2]

V = kq/|z|

Here we have employed q = πσR2 for the net charge on the disk. This is similar as the potential due to the point charge.

Electrostatic Potential Energy:

Electric potential energy or simply electrostatic potential energy is a potential energy (computed in joules) that outcomes from conservative Coulomb forces and is related by the configuration of a specific set of point charges in a defined system. An object might encompass electric potential energy through virtue of two key elements: its own electric charge and its relative position to the other electrically charged objects.

The word 'electric potential energy' is employed to explain the potential energy in systems having time-variant electric fields, whereas the word 'electrostatic potential energy' is employed to explain the potential energy in systems having time-invariant electric fields.

Define: The electrostatic potential energy can be stated in terms of electric field or in terms of the electric potential. Both the given definitions are entirely valid and can be employed equally.

The electrostatic potential energy, UE, of one point charge 'q' at position 'r' in the presence of an electric field 'E' is stated as the negative of the work done 'W' by means of the electrostatic force to bring it from the reference position rref to that position 'r'.

UE(r) = -Wref→r = - r ref ∫r qE.ds

Here 'E' is the electrostatic field and ds are the displacement vector in a curve from the reference position rref to the final position 'r'.

The electrostatic potential energy can as well be stated from the electric potential as shown:

The electrostatic potential energy, UE, of one point charge 'q' at position 'r' in the presence of an electric potential Φ is stated as the product of the charge and the electric potential.

UE(r) = q Φ (r)

Here Φ is the electric potential produced through the charges that is a function of position 'r'.

Work done 'W' is moving  a charge 'q' from point A to a point B in the area of the electric field 'E' is:

W = - A∫B F.dr = - q A∫B E. dr     

Here 'F' is the electrostatic force on q. We as well know that the line integral of the electric filed, that is, A∫B E. dr is independent of the path between A and B. This means that the line integral of the electrostatic force, that is A∫B E. dr is as well independent of the path between A and B.

In another words, the work done on a charged particle in moving it against the electrostatic force 'E' is independent of the path between A and B, and based merely on the point A and B.

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