Potential, Potential Difference & Potential Energy – Definitions, Work & Conservation

TAKEAWAYS:

• Electric potential is the work done per unit charge to bring a charge from infinity to a point in an electric field.
• Potential difference (V) between two points is the work done per unit charge to move a charge between them.
• Potential energy (U) of a charge in an electric field depends on its position and the field.
• The conservation of energy in electric fields means that the total energy (kinetic + potential) remains constant if no work is done by external forces.
• Potential difference is measured in volts (V), and potential energy is measured in joules (J).
• The electric potential due to a point charge follows the inverse-square law, just like gravitational potential.

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What is Electric Potential?

Electric potential at a point in an electric field is defined as the work done per unit positive charge to bring a charge from infinity to that point. It is a scalar quantity (has only magnitude, no direction).

Key Points:

• Symbol: $V$ (or $V$)
• Unit: Volt (V), named after Alessandro Volta.
• Formula: $$V=\frac{W}{q}$$ where $W$ is the work done and $q$ is the test charge.

Visualizing Potential:

Imagine a positive charge placed in space. The electric field around it exerts a force on other charges. To bring another positive charge closer to it, you must do work against the repulsive force. The electric potential at a point is like the "height" of that point in an electric "hill." The higher the potential, the more work you need to do to bring a charge there.

This graph shows how electric potential decreases as you move farther from a point charge. The potential is highest near the charge and drops off rapidly.

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Potential Difference (Voltage)

Potential difference (PD) between two points is the work done per unit charge to move a charge from one point to another in an electric field. It is also called voltage.

Key Points:

• Symbol: $V_{AB}$ (potential difference between points A and B)
• Unit: Volt (V)
• Formula: $$V_{AB}=\frac{W_{AB}}{q}=V_B-V_A$$ where $W_{AB}$ is the work done to move charge $q$ from A to B.

Example:

Suppose you move a charge of $2\text{C}$ between two points A and B, and the work done is $10\text{J}$. The potential difference between A and B is: $$V_{AB}=\frac{10\text{J}}{2\text{C}}=5\text{V}$$

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Relationship Between Electric Field and Potential

The electric field ($E$) and electric potential ($V$) are related. For a uniform electric field (like between two parallel plates), the potential difference is given by: $$V=E\cdot d$$ where:

• $E$ = electric field strength (N/C)
• $d$ = distance between the two points (m)

Worked Example:

A uniform electric field of $500\text{N/C}$ exists between two parallel plates separated by $0.1\text{m}$. Calculate the potential difference between the plates. $$V=E\cdot d=500\text{N/C}\times 0.1\text{m}=50\text{V}$$

In a uniform electric field, the potential decreases linearly from the positive plate to the negative plate.

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Potential Due to a Point Charge

The electric potential $V$ at a distance $r$ from a point charge $Q$ is given by: $$V=\frac{kQ}{r}$$ where:

• $k=9\times 10^9{\text{N m}}^2/{\text{C}}^2$ (Coulomb’s constant)
• $Q$ = charge (C)
• $r$ = distance from the charge (m)

Worked Example:

Calculate the electric potential at a point $0.5\text{m}$ away from a charge of $2\mu \text{C}$ ($2\times 10^{-6}\text{C}$). $$V=\frac{(9\times 10^9)(2\times 10^{-6})}{0.5}=36,000\text{V}=36\text{kV}$$

This graph shows how potential decreases as distance from the charge increases.

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Potential Energy of a Charge in an Electric Field

The electric potential energy (U) of a charge $q$ at a point in an electric field is the work done to bring it from infinity to that point. It is given by: $$U=qV$$ where:

• $U$ = potential energy (J)
• $q$ = charge (C)
• $V$ = electric potential (V)

Worked Example:

A charge of $3\text{C}$ is placed at a point where the electric potential is $10\text{V}$. Calculate its potential energy. $$U=qV=3\text{C}\times 10\text{V}=30\text{J}$$

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Conservation of Energy in Electric Fields

In an electric field, the total energy (kinetic + potential) of a charge remains constant if no external work is done. This is similar to conservation of mechanical energy in gravity.

Key Idea:

• If a charge moves from a point of high potential to low potential, its potential energy decreases, and its kinetic energy increases (and vice versa).
• Mathematically: $$K_i+U_i=K_f+U_f$$ where $K$ = kinetic energy, $U$ = potential energy, and subscripts $i$ and $f$ denote initial and final states.

Worked Example:

A charge of $1\text{C}$ moves from a point at $20\text{V}$ to a point at $10\text{V}$. If its initial kinetic energy is $5\text{J}$, what is its final kinetic energy? $$U_i=q{V}_i=1\times 20=20\text{J}$$ $$U_f=q{V}_f=1\times 10=10\text{J}$$ Using conservation of energy: $$5+20=K_f+10⟹K_f=15\text{J}$$

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Comparison Table: Potential vs. Potential Difference vs. Potential Energy

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Applications of Potential and Potential Difference

1. Batteries: Provide a potential difference to drive current in circuits.
2. Capacitors: Store energy in the form of electric potential.
3. Electric Power Transmission: High-voltage transmission reduces energy loss.
4. Electrostatic Precipitators: Used in industries to remove pollutants from smoke.
5. Medical Equipment: Defibrillators use high voltage to restart hearts.

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NEB Board-Style Questions

Short Answer Questions

1. Define electric potential. How is it related to work done?
2. A charge of $5\mu \text{C}$ is moved between two points with a potential difference of $100\text{V}$. Calculate the work done.
3. What is the potential difference between two points if $10\text{J}$ of work is done to move $2\text{C}$ of charge between them?
4. Explain why electric potential is a scalar quantity, while electric field is a vector quantity.

Long Answer Questions

5. Derive the expression for the electric potential due to a point charge. How does it vary with distance?
6. A uniform electric field of $200\text{N/C}$ exists between two parallel plates separated by $0.05\text{m}$. Calculate:
   • The potential difference between the plates.
   • The work done to move a charge of $0.1\text{C}$ from the negative to the positive plate.
7. Explain the principle of conservation of energy in an electric field. A charge of $3\text{C}$ moves from a point at $15\text{V}$ to a point at $5\text{V}$. If its initial kinetic energy is $10\text{J}$, find its final kinetic energy.

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Exam Tip

1. Understand the Definitions: Always remember that potential is work per unit charge, and potential difference is the difference in potential between two points.
2. Units Matter: Potential is in volts (V), potential energy in joules (J). Never mix them up!
3. Sign Conventions: Potential is higher near positive charges and lower near negative charges. Potential difference is $V_{AB}=V_B-V_A$.
4. Conservation of Energy: In problems involving motion of charges, always check if energy is conserved. Use $K_i+U_i=K_f+U_f$.
5. Practical Applications: Know how potential difference is used in real-life devices like batteries, capacitors, and power transmission.
6. Graphs and Diagrams: Draw diagrams to visualize electric fields and potential variations. Label points clearly.
7. Numerical Problems: Practice calculating work done, potential difference, and potential energy using the given formulas. Always show your steps.

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Electric potential lines (red) and equipotential surfaces (blue) around a positive charge. (Image: Balajijagadesh, CC BY-SA 4.0, via Wikimedia Commons) Uniform electric field between two parallel plates with potential difference. (Image: Geek3, CC BY-SA 4.0, via Wikimedia Commons)