Electric Potential Energy and Electric Potential
Electric potential energy and electric potential are fundamental concepts in understanding how charged particles interact.
Electric Potential Energy
Electric potential energy
Electric potential energy is the energy stored in a system of charged particles due to their positions relative to each other.
For two point charges, the electric potential energy is given by:
$$
E_p = \frac{k q_1 q_2}{r}
$$
where:
- $E_p$ is the electric potential energy.
- $k$ is the Coulomb constant ($8.99 \times 10^9 \, \text{N m}^2/ \text{C}^2$).
- $q_1$ and $q_2$ are the magnitudes of the charges.
- $r$ is the distance between the charges.
The formula resembles Coulomb’s law, but instead of force, it calculates energy.
Properties of Electric Potential Energy
- Scalar Quantity: Electric potential energy is a scalar, meaning it has magnitude but no direction.
- Depends on Charge Signs:
- If the charges are of opposite signs, the energy is negative, indicating an attractive interaction.
- If the charges have the same sign, the energy is positive, indicating a repulsive interaction.
- Reference Point: The reference point for electric potential energy is usually taken at infinity, where the energy is zero.
- Consider two charges,$q_1 = 2 \, \mu\text{C}$ and $q_2 = -3 \, \mu\text{C}$, separated by $0.05 \, \text{m}$.
- The electric potential energy is: $$
E_p = \frac{(8.99 \times 10^9) (2 \times 10^{-6})(-3 \times 10^{-6})}{0.05} = -1.08 \, \text{J}
$$ - The negative sign indicates an attractive interaction.
Electric Potential
Electric potential
Electric potential is the amount of work done per unit charge in bringing a small positive test charge from infinity to a point in an electric field.
It is defined as:
$$
V_e = \frac{k Q}{r}
$$
where:
- $V_e$ is the electric potential.
- $Q$ is the charge creating the potential.
- $r$ is the distance from the charge.
The unit of electric potential is the volt (V), where $1 \, \text{V} = 1 \, \text{J C}^{-1}$.
Properties of Electric Potential
- Scalar Quantity:
- Like electric potential energy, electric potential is a scalar.
- Independent of Test Charge:
- The potential depends only on the charge creating the field, not on the test charge.
- Zero at Infinity:
- The potential is often defined to be zero at infinity.
A charge $Q = 5 \, \mu\text{C}$ creates an electric potential at a point $0.1 \, \text{m}$ away:
$$
V_e = \frac{(8.99 \times 10^9)(5 \times 10^{-6})}{0.1} = 449,500 \, \text{V}
$$
Relationship Between Electric Potential and Electric Potential Energy
The electric potential energy $E_p$ of a charge $q$ at a point with electric potential $V_e$ is given by:
$$
E_p = q V_e
$$
If a charge $q = 2 \, \mu\text{C}$ is placed at a point where the electric potential is $500 \, \text{V}$, the electric potential energy is:
$$
E_p = (2 \times 10^{-6})(500) = 0.001 \, \text{J}
$$
Field Strength as Potential Gradient
- The electric field strength $E$ is related to the potential difference $\Delta V_e$ by the formula: $$
E = -\frac{\Delta V_e}{\Delta r}
$$ - This equation shows that the electric field is the rate of change of electric potential with distance.
The negative sign indicates that the electric field points in the direction of decreasing potential.
- If the potential decreases by $100 \, \text{V}$ over a distance of $0.5 \, \text{m}$, the electric field strength is: $$
E = -\frac{100}{0.5} = -200 \, \text{V m}^{-1}
$$ - The negative sign indicates the field points in the direction of decreasing potential.
Work in an Electric Field
- Work is done when a charge moves in an electric field.
- The work done $W$ in moving a charge $q$ through a potential difference $\Delta V_e$ is given by: $$
W = q \Delta V_e
$$
If a charge $q = 3 \, \mu\text{C}$ moves through a potential difference of $200 \, \text{V}$, the work done is:
$$
W = (3 \times 10^{-6})(200) = 0.0006 \, \text{J}
$$
- Students often confuse electric potential with electric potential energy.
- Remember, potential is energy per unit charge, while potential energy is the total energy for a specific charge.
- What is the electric potential energy between two charges of $+2 \, \mu\text{C}$ and $-3 \, \mu\text{C}$ separated by $0.1 \, \text{m}$?
- How much work is done in moving a $5 \, \mu\text{C}$ charge through a potential difference of $150 \, \text{V}$?
- If the electric potential at a point is $300 \, \text{V}$, what is the electric potential energy of a $4 \, \mu\text{C}$ charge at that point?
