Introduction
Electric potential energy is the process that happens due to two elements-one which is possessed by the object itself, the other is the relative position of the object. The result of the electric potential completely depends on the total work done in moving the object from one point to another.
Imagine you have a negative charged plate, with a little positive charged particle stuck to it through the electric force. You will witness an electric field around the plate pulling positively charged objects towards it. Now take the positive particle, and pull it off the plate against the electric field. This could be hard work because the electric force is pulling all together. The energy used for moving the particle from the plate is stored in the particle as the electrical potential energy.
What is Electrical Potential Energy?
To understand about the deriving electric field from potential, it is important to know the meaning of the electrical potential energy. Electric potential energy of the given charge or system of changes is termed as the total work done by the external agent to bring the charge or the system of charges. In other words, electric potential energy is defined as the total potential energy a unit charge will possess if located at any point in the exterior space.
Overview
Electric potential energy is scalar quantity and possesses only magnitude and no direction.
Electric Potential Formula:
Charge placed in an electric field possesses potential energy and is measured by the work done in moving the charge from infinity to the point against the electric field. If two charges q1 and q2 are separated by the distance D, electric potential energy of the system is- U = 1/ (4πεo) × [q1q2/d].
What is Electric Potential?
Electric potential or voltage is the difference in potential energy per unit charge between two locations in an electric field. It is important to know that the amount of charge you are pushing or pulling makes a huge difference to the electrical potential energy. So is why physicists use single positive charge as our imaginary charge to test out the electrical potential.
Electric Potential Derivation:
To understand this, you need to consider a charge q1. Let's say, they are placed at the distance "r" from each other. Total electric potential of the charge is defined as the total work done by an external force.
We can write it as, -∫ (ra → rb) F.dr = – (Ua – Ub)
Here, we see that the point rb is present at infinity and the point ra is r.
Substituting the values we can write, -∫ (r →∞) F.dr = – (Ur – U∞)
As we know that Infinity is equal to zero.
Therefore, -∫ (r →∞) F.dr = -UR
Using Coulomb’s law, between the two charges we can write:
⇒ -∫ (r →∞) [-kqqo]/r2 dr = -UR
Or, -k × qqo × [1/r] = UR
Therefore, UR = -kqqo/r
What is Meant By Electric Potential Difference?
In an electrical circuit, potential between two points (E) is defined as the amount of work done (W) by the external agent in moving the unit charge.
In mathematical way we can say that: E = W/Q.
E = Electrical potential difference between two points.
W = Work done in moving a change from one point to another.
Q = Quantity of charge in coulombs.
FAQs on Deriving Electric Field From Potential
1. What is the fundamental relationship between electric field and electric potential?
The fundamental relationship is that the electric field (E) at any point is equal to the negative of the potential gradient at that point. In simpler terms, the electric field measures the rate at which the electric potential (V) changes with distance. The formula for this relationship in one dimension is E = -dV/dr.
2. How do you derive the formula relating electric field to potential for a Class 12 student?
As per the CBSE 2025-26 syllabus, the derivation involves considering the work done in moving a small positive test charge (q₀) through a small distance (dr) against an electric field (E).
The steps are as follows:
- The work done (dW) by an external force to move the charge is: dW = F_ext ⋅ dr. To move it without acceleration, the external force must be equal and opposite to the electric force (F_e = q₀E), so F_ext = -q₀E.
- Substituting this, we get: dW = -q₀E ⋅ dr.
- By definition, the change in electric potential (dV) is the work done per unit charge: dV = dW / q₀.
- Combining these two equations: dV = (-q₀E ⋅ dr) / q₀, which simplifies to dV = -E ⋅ dr.
- Therefore, the electric field component along dr is given by E = -dV/dr.
3. What is the physical significance of the negative sign in the equation E = -dV/dr?
The negative sign is crucial and signifies that the direction of the electric field (E) is always in the direction of the steepest decrease in electric potential (V). A positive charge naturally experiences a force that moves it from a region of higher potential to a region of lower potential. The electric field vector points along this path of decreasing potential.
4. How can you determine the electric field from a diagram of equipotential surfaces?
You can determine the electric field from a diagram of equipotential surfaces by observing two key properties:
- Direction: The electric field lines are always perpendicular to the equipotential surfaces at every point. They point from a surface of higher potential to a surface of lower potential.
- Magnitude: The magnitude of the electric field is inversely proportional to the spacing between the equipotential surfaces. Where the surfaces are closer together, the field is stronger. Where they are farther apart, the field is weaker.
5. Is it possible for the electric potential to be non-zero at a point where the electric field is zero?
Yes, this is possible. A classic example is the space inside a charged hollow spherical conductor. The electric field (E) is zero everywhere inside the cavity. However, the electric potential (V) is constant and non-zero, having the same value as the potential on the surface of the conductor. This happens because there is no change in potential (dV/dr = 0) inside, which corresponds to E=0.
6. Conversely, can the electric field be non-zero where the electric potential is zero?
Yes, the electric field can be non-zero at a point where the electric potential is zero. A common example is the equatorial point of an electric dipole. At the midpoint on the line perpendicular to the dipole axis, the potentials from the positive and negative charges cancel each other out, making the net potential V = 0. However, the electric field vectors from both charges have components that add up, resulting in a non-zero net electric field.
7. How is the relationship between electric field and potential expressed in three dimensions?
In three dimensions, the electric field (E) is expressed as the negative gradient of the scalar potential field (V), using partial derivatives with respect to the x, y, and z coordinates. The relationship is given by the vector equation:
E = - ( î ∂V/∂x + ĵ ∂V/∂y + k̂ ∂V/∂z )
Here, î, ĵ, and k̂ are the unit vectors along the x, y, and z axes, respectively. This equation shows that each component of the electric field (Ex, Ey, Ez) is the negative partial derivative of the potential with respect to its corresponding coordinate.
