Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

More Examples  of Electric Field and Potential

Example 1: Two intersecting and oppositely charged spheres :

We consider the electric field in the region of intersection of two oppositely charged but otherwise identical spheres. The spheres are assumed to be charged uniformly.

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We have seen that the electric field inside a uniformly charged sphere is linear in r, the distance from the centre,

  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the region of intersection is interior to both the spheres, at a point P in this region, we have,

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

It is seen that the field depends only on the distance between the two centres and is independent of position of P within this intersection.

Example 2 : Field inside a non-concentric cavity inside a uniformly charged sphere

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Consider a sphere of radius R with its centre at O. There is an off centre cavity of radius a whosecentre is at O’.

The way to do these problems is to consider the cavity as a superposition of equal and oppositely charged spheres , the charge density of the two spheres being equal to the charge density of the charged part of the bigger sphere. This makes the problem equivalent to finding the field inside the cavity region due to a sphere of charge density with its centre at O and another sphere of charge density  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) with its centre at O’ 

Let the point P be at a position  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) respect to Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) then the field at P due to the bigger sphere (with density Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) while that due to the smaller sphere of charge density Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the net field is given by Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  which is constant inside the cavity.

Nature of Coulomb Force:

1. It is a central force, inverse square in nature.
2. Force is conservative : 

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

3. Since the force is conservative, it can be expressed as a gradient of a scalar potential,  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is known as scalar potential.

Electrostatic Potential :

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Using Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The gradient operator can be taken outside as the gradient is with respect to unprimed variable Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) ,while the integration is with respect to primed variable. This gives,

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Note that the divergence of the electric field can be written as

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

TheLaplacian operator, acting on unprimed variable can be taken inside the integration,

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Thus we have the following two relations for the electrostatic field :

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Suppose you are bringing a unit charge from some reference point (where the potential is defined as zero), the work that needs to be done by you (i.e. by an external agency) to bring this charge from the reference point to a position Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)is negative of the work done by the force on the charge. The work done by the force is
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

For instance, for field produced by a point charge q at the origin, the potential at the point Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  is (where the reference point is taken to be at infinity)
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
The name “potential” is very similar to the phrase “potential energy” and this is often confusing because though there is a connection between the two, they are different things. The relationship is understood by considering the work that needs to be done to bring a test charge q from the reference point to the point P where it is to be placed, The work done by an external agency, as calculated above is Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) This work done then becomes the potential energy of the system. Thus the potential at a point can be interpreted as the potential energy associated with a unit point charge at that point.

Poisson’s Equation

We have seen that the divergence of the electric field satisfies Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Substituting  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) in this equation, we get Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

This is known as the Poisson’s equation. In a region of space where there are no charges the equation satisfied by the potential is the Laplace’s equationExamples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
 

Example 3: Potential due to a line charge

We have seen that the field due to a line charge can be evaluated by enclosing the line charge with a Gaussian cylinder of length l and radius r. The field can only depend on the distance from the wire and be directed away from the wire. The contribution to the flux from the top and bottom caps are zero and the flux is contributed only by the curved surface of the cylinder, giving,

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which gives (plugging in the direction of the field) Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Equating this to the gradient of potential, (which in this case is just the derivative with respect to r) we get,
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Unlike in the case of a point charge, the reference point of the potential cannot be taken at infinite distances because logarithm gets undefined there. We can choose some arbitrary unit and choose the potential to be zero at r=1 and with respect to this reference point, the potential at a distance r from the line charge is given by

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Example 4 : Screened Coulomb Potential

A screened Coulomb potential, also known as Yukawa potential arises in systems such as semiconductors where the medium in which interaction takes place (a dielectric) partially screens the bare Coulomb interaction between charges. The potential is given by

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
The charge distribution that gives rise to this electric field can be obtained by calculating the divergence of the electric field
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which shows that there is a point charge at the origin in addition to an exponentially decaying charge density. The total charge can be obtained by integrating the charge density over the entire volume

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

What is the advantage of using the potential formulation over calculating electric field? Potential being a scalar is easier to deal with mathematically. When dealing with multiple charges, it is simpler to add up the potential using superposition principle and then take the gradient to determine the electric field. Potential is analogous to pressure in a fluid. Just as a fluid in a pipe tends to move from a region of high pressure to that of low pressure, in an electric field positive charges tend to move from a region of higher potential to that of a lower potential.

Potential and Electric Field of a dipole

A dipole is basically two equal and opposite charges separated by a distance. For an ideal dipole the separation goes to zero. Dipole moment is a vector defined as a vector of magnitude equal to the product of the magnitude of either charge and the distance between them. The direction of the dipole moment vector is defined as along the direction from the negative charge to the positive charge.

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

In the figure the positive charge is shown by the blue circle and the negative charge by the blue. The dipole moment vector is given by  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where d is the distance between the charges. From simple geometry, it can be seen that,

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Thus

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

so that the potential at a point P is given by

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The electric field can be calculated from a knowledge of the potential. The gradient operator in spherical polar is

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The geometry is shown in the figure. It is seen that the unit vector along the dipole moment (taken along the z axis) can be written as,  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) so that,

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

This is the dipole field in a coordinate independent form. The lines of force due to a dipole is shown below :

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Tutorial Assignment

1. We have seen that the potential due to a point charge located at the origin at the position Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where the zero of the potential is taken at infinity. How would this expression change if the zero of the potential was taken on a sphere of radius R about the origin?

2. The electric field in a cubical region of space  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is given by the following expression : Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

verify that this expression can represent an electrostatic field and determine the charge density and the total charge that gives rise to this electric field.

3. A spherical charge distribution is given by Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and zero outside this region. Obtain a potential corresponding to this distribution.

Solutions to Tutorial Assignment

1. We have to take the line integral of the electric field from r=R to the point in question   Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

2. It can be verified that the curl of the electric field is zero so that the given expression represents a valid electrostatic field. The divergence of the field is given by  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the total charge is
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
3. The electric field is spherically symmetric.  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) which gives Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) (There is an additional tem  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) possible but we take C=0 so that field does not diverge at r=0. Alternatively, you can use Gauss’s law to arrive at the correct expression for the electric field). The potential corresponding to this field is  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the charge density being zero, the potential is Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Continuity of the potential at r=R gives , for r < R
Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Self Assessment Quiz

1. Verify if the following field can represent an electrostatic field in a given region.

Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

If yes, find a potential function and determine the charge density at the point (2,3,0).

2. The electric field in certain region of space is given in spherical polar coordinates as  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Determine the charge density.

3. Find the electric field of a dipole on a point (a) along its axis and (b) along its perpendicular bisector in Cartesian coordinates.

4. Show that the magnitude of the electric field of a dipole at an angle θ to the axis of the dipole is given by  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Solutions to Self Assessment Quiz

1. The curl of the field is zero and hence the field can be electrostatic. The potential function is  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The divergence of the field is given by 10x + 12yz which is equal to  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus at (2,3,0) the charge density is 20∈0.

2. Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
3. We have shown that the field of an electric dipole at an angle θ to the axis of the dipole is given by Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Taking the axis along the z axis θ = 0, along the axis so that the field is  Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Perpendicular to the axis (take it as x direction, Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the field is Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

4. This can be solved in a straightforward manner.

The document Examples: Electric Field & Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Electromagnetic Fields Theory (EMFT).
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FAQs on Examples: Electric Field & Potential - Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

1. What is an electric field?
Ans. An electric field is a region around a charged object where other charged objects experience a force. It is created by electric charges and can be either positive or negative. The strength and direction of the electric field are determined by the magnitude and distribution of the charges.
2. How is the electric field calculated?
Ans. The electric field at a point is calculated by dividing the force experienced by a test charge placed at that point by the magnitude of the test charge itself. Mathematically, the electric field (E) is equal to the force (F) divided by the test charge (q): E = F/q.
3. What is electric potential?
Ans. Electric potential, also known as voltage, is the amount of electric potential energy per unit charge at a given point in an electric field. It represents the work done to move a unit positive charge from infinity to that point in the electric field. Electric potential is measured in volts (V).
4. How is electric potential different from electric field?
Ans. While electric field represents the force per unit charge experienced by a charged object, electric potential represents the electric potential energy per unit charge at a given point. Electric field is a vector quantity, having both magnitude and direction, whereas electric potential is a scalar quantity, having only magnitude.
5. Can electric potential be negative?
Ans. Yes, electric potential can be negative. The sign of the electric potential depends on the nature of the charge creating the field. If the charge creating the field is positive, the electric potential at a point will be positive. However, if the charge creating the field is negative, the electric potential at a point will be negative.
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