Potential Difference

Electrical charges need energy to push them along a circuit. Water always flows from higher to lower ground; similarly, an electric charge always flows from a point at higher potential to a point at lower potential. An electric current can flow only when there is a potential difference (V) or p.d.

The potential difference (p.d.) between two points is equal to the work done per unit charge in moving a positive test charge from one point to the other.

OR

The work done per unit positive charge in moving a charge from one point to another in an electric field is called the potential difference between those two points.

OR

The potential difference between any two points in the circuit is the amount of energy needed to move one unit of electric charge from one point to the other.

\[V_A-V_B=\frac{W}{Q}\]

The potential difference between two points is said to be 1 volt if the work done in moving 1 coulomb charge from one point to the other is 1 joule:

1 volt = \[\frac {\text {1 joule}}{\text {1 coulomb}}\] = 1 J C⁻¹

1. Aim: To observe the flow of electric current in a circuit and understand the role of potential difference.

2. Requirements: copper connecting wires, a light bulb, and a 1.5 V dry cell (battery).

3. Procedure

• Build a circuit with copper wires and a light bulb as shown in Fig. a. The bulb does not light up because there is no current.
• Connect a 1.5 V dry cell as in Fig. b. The bulb lights up, indicating current flow.
• Electrons move from the negative to the positive terminal due to the potential difference created by the dry cell.
• Conventional current flows in the opposite direction of electron movement.

(a) Electrical Circuit

 (b) Electrical Circuit

4. Conclusion: The bulb does not light up in the circuit without a cell because there is no potential difference to drive the current. When a cell is connected, a potential difference is applied, causing current to flow and the bulb to light up. The SI unit of potential difference is the volt (V), and the SI unit of electric current is the ampere (A), which is defined as 1 coulomb of charge passing through a wire per second.

Derivation from Potential Energy

• The gravitational potential energy (U) of a system containing Earth (mass M) and an object (mass m) at distance r is given by:
  U = -G\[\frac {Mm}{r}\]
• We can rearrange this equation to separate the terms:
  U = \[\left( - \frac{GM}{r} \right)\]m
• The term in the bracket \[\left( - \frac{GM}{r} \right)\] is identified as the Gravitational Potential (V_E) of Earth.
• This factor depends only on:
  The mass of the Earth (M).
  The location/distance from the center (r).

Relationship between Potential and Energy

Using the definition above, we can write the relationship as:
U = V_r × m

• Where V_r is the gravitational potential at distance r.
• Alternatively, \[V_r = \frac{U}{m}\].

Gravitational Potential Difference

• The difference in potential between any two points in a gravitational field relates to the work done.
• It is defined as the change in potential energy per unit mass or work done (dW) per unit mass.
  V2 − V1 = \[\frac{U_{2}-U_{1}}{m}\] = \[\frac {dW}{m}\]

General Case for Two Masses

For any two masses m₁ and m₂ separated by distance r, the potential energy can be interpreted in two ways:

1. Potential of m₁ at location r (V₁) multiplied by mass m₂.
2. Potential of m₂ at location $r$ (V₂) multiplied by mass m₁.

• Formula:
  U = \[-G\frac{m_1m_2}{r}\] = (V₁)m₂ = (V₂)m1

1. Aim: To demonstrate how a potential difference causes the flow of charges between two points.

2. Requirements: two conductors (A and B), a conducting wire, and insulated stands for safety.

3. Procedure

• Place conductor A at a higher potential (positive charge) and conductor B at a lower potential (negative charge).
• Connect the two conductors using a conducting wire.
• Observe the flow of electrons from B (lower potential) to A (higher potential).

Potential difference and flow of electricity

4. Observation: Electrons flow from conductor B to conductor A because of the potential difference. The flow of electrons continues until the potential difference between A and B becomes zero.

5. Conclusion: A potential difference between two points drives the flow of charges. Positive charges move from higher to lower potential, while electrons (negative charges) move from lower to higher potential. Work is required to move a positive charge against the electric field, from lower to higher potential. This experiment explains how potential difference causes electric current in a circuit.

• Conservative Nature: The electric field is conservative, meaning the work done in moving a charge between two points is path-independent and depends only on the potential difference.
• Potential Difference and Energy Storage: The potential difference (V_A − V_B) determines how much energy is transferred per unit charge between two points, fundamental in circuits and electrostatic potential energy calculations.
• Equipotential Surfaces: Surfaces where the potential remains constant, meaning no work is required to move a charge along them. The electric field is always perpendicular to equipotential surfaces.

Problem 5

A charge of 2 × 10⁴ C flows through an electric heater. The amount of electrical energy converted into thermal energy is 5 × 10⁶ J. Compute the potential difference across the ends of the heater.

Solution:

Two points A and B are 2 cm apart and a uniform electric field \[\vec E\] acts along the straight line AB directed from A to B with E = 200 N-C⁻¹. A particle of charge +10⁻⁶ C is taken from A to B along AB. Calculate: (a) the force on the charge, (b) the potential difference (V_A − V_B), and (c) the work done on the charge by \[\vec E\].

Solution:

1. \[\vec F\] = q\[\vec E\] = 10⁻⁶ C × 200 N-C⁻¹ = 2 × 10⁻⁴ N along AB.
2. V_A − V_B = Ed = 200 N-C⁻¹ × 0.02 m = 4 V.
3. W = \[\vec \] . \[\vec d\] = (2 × 10⁻⁴ N)(0.02 m) = 4 × 10⁻⁶ J.