What is Potential Energy?
Potential energy is the energy that results from the position/configuration of a body. It can come from its position relative to others, internal stress, electrical charge, or its condition. Owing to its position in a gravitational field (gravitational potential energy), an electric field (electrical potential energy), or a magnetic field (magnetic potential energy), an object may have the capacity to do work. A stretched spring or other elastic deformation can have elastic potential strength.
Potential Energy Function
If a force acting on an object is only a function of position, it is said to be a conservative force and can be represented by a potential energy function that satisfies the derivative condition for a one-dimensional case.
Fx = −dU/dx
After integrating this equation,
U(x) = - ∫ F(x)dx
This is the definition of potential energy. There is an arbitrary integration constant in the above equation, which shows that any constant can be added to the potential energy equation. This means that you can set the potential energy to zero at any point, which is convenient.
Zero Potential
The nature of potential is such that the zero potential is arbitrary; it can be set as the center of a coordinate system or at the infinity to the system. This is not to suggest it's insignificant. Once the potential zero is set, then every possible value is evaluated with respect to that zero. The zero of electrical potential has been set for convenience, but the choice of this zero point is generally based on some physical or geometrical logic. It is appropriate to set the zero point at infinity for a single point charge or the localized array of charges. But for an infinite line charge, that's not a rational option, since local values of the potential will go to infinity. The earth or ground potential is usually assumed to be zero for functional electrical circuits, and everything is referred to as the earth.
The Potential Energy of a Single Charge in an Electric Field
Let us consider a magnitude q charge put in a magnitude E, which is very small. The electrical potential at a point is equal to the potential electrical energy (Joules) of any charged particle at that position, divided by the particle charge (Coulombs). Here, the E is the external source of the electric field, which might be unknown as well. This external field is not affected by the charge q. In the field, the potential energy of the charge q is equal to the work done to bring the charge from infinity to the point. The potential at infinity is always assumed to be zero; the work performed to bring the charge from infinity to point is assigned as qV.
The potential energy of a single charge is given by,
qV(r).
Where, r is the position vector, and V(r) is external potential at point r.
The Potential Energy of the System of Two Charges in an Electric Field
Let us consider two charges with magnitude q1 and q2. They are at a distance of r1 and r2 from the origin for the respective charge. Both the charges are placed in the external field of magnitude E. Work done on charge q1 while bringing it from infinity to r1 is q1V(r1), and for the charge, q2 is q2V(r2). Now, in this case, there are two charges, so there will be an electric potential caused due to a charge as well. So for the charge q2, there will be a field due to the q1 along with the electric field E which will be given by
(q1q2)/(4πϵ0r12)
r12 is the distance between the charges q1 and q2.
Now, work done for the charge q2 will be given by:
q2V(r2) + (q1q2)/(4πϵ0r12)
Now, the total potential energy will be given by:
q1V(r1) + q2V(r2) + (q1q2)/(4πϵ0r12)
Where, r1 and r2 are the position vector and V(r1), and V(r2) is the external potential at point r1 and r2, respectively.
Fun Facts
Here are some fun facts to learn in relation to potential energy in an external field in class 12:
The electrical potential V is a scalar and has no direction, whereas the electric field E is a vector.
To possess potential energy, an entity must be modified by some form of force.
The heavier an object and the higher it is above the ground, the more energy it has in its gravitational potential.
FAQs on Potential Energy of Charges in an Electric Field
1. What is meant by the potential energy of a charge in an electric field?
The electric potential energy of a charge at a point in an electric field is defined as the total amount of work done by an external force in bringing that charge from infinity to that specific point, without causing any acceleration. It represents the energy stored in the charge by virtue of its position in the field.
2. What is the formula for the electric potential energy of a system of two point charges?
The electric potential energy (U) for a system of two point charges, q₁ and q₂, separated by a distance 'r' in a vacuum is given by the formula:
U = (1/4πε₀) * (q₁q₂ / r)
Here, 1/4πε₀ is Coulomb's constant (k ≈ 9 × 10⁹ N m²/C²), q₁ and q₂ are the magnitudes of the charges, and r is the distance between them.
3. How does electric potential energy differ from electric potential?
These two concepts are related but distinct. The key difference is:
- Electric Potential (V) is a characteristic of a point in an electric field. It is the work done per unit charge (V = W/q) to bring a charge to that point. Its unit is Volts (V).
- Electric Potential Energy (U) is the energy that a specific charge possesses due to its position in that field. It is the product of the charge and the electric potential at its location (U = qV). Its unit is Joules (J).
In simple terms, potential is a property of the location, while potential energy is a property of the charge placed at that location.
4. Can a single, isolated charge possess electric potential energy?
No, a single, isolated charge in empty space does not have electric potential energy by itself. Potential energy arises from the interaction between two or more charges. However, if a single charge is placed in an external electric field (a field created by other, separate charges), it will possess potential energy determined by its position within that external field.
5. What is an electron-volt (eV) and why is it a useful unit for potential energy?
An electron-volt (eV) is a unit of energy. It is defined as the amount of kinetic energy gained by a single electron when it is accelerated through an electric potential difference of one volt. It is a much smaller unit than the Joule (1 eV ≈ 1.602 × 10⁻¹⁹ J). This unit is commonly used in atomic and nuclear physics because the energies involved at the subatomic level are extremely small and more conveniently expressed in eV.
6. How is the potential energy calculated for a system of two charges placed in an external electric field?
The total potential energy of a system of two charges (q₁ and q₂) in an external electric field is the sum of three components:
- The potential energy of the first charge in the external field: U₁ = q₁V(r₁), where V(r₁) is the potential of the external field at the position of q₁.
- The potential energy of the second charge in the external field: U₂ = q₂V(r₂), where V(r₂) is the potential of the external field at the position of q₂.
- The mutual potential energy due to their interaction with each other: U₁₂ = (1/4πε₀) * (q₁q₂ / r₁₂), where r₁₂ is the distance between them.
The total potential energy is U_total = q₁V(r₁) + q₂V(r₂) + (1/4πε₀) * (q₁q₂ / r₁₂).
7. What is the physical significance of positive and negative electric potential energy?
The sign of the electric potential energy indicates the nature of the interaction between charges:
- Negative Potential Energy: This occurs between opposite charges (one positive, one negative). It signifies an attractive force. The system is stable, and external work must be done to separate the charges.
- Positive Potential Energy: This occurs between like charges (both positive or both negative). It signifies a repulsive force. The system is unstable, and the charges will fly apart if released, converting their potential energy into kinetic energy.
