**Force between Parallel Currents:** Definition of one Ampere and force between two parallel current-carrying conductors

📌 1. Question Description

Force Between Parallel Current-Carrying Conductors

When electric currents flow through two long, straight and parallel conductors, each conductor produces a magnetic field around itself. This magnetic field exerts a force on the current-carrying conductor placed in it.

The force per unit length between two parallel conductors is given by:

F/L = μ₀ I₁ I₂ / 2πd

Where:

• I₁ and I₂ = currents in the two conductors
• d = distance between the conductors
• μ₀ = permeability of free space
• F = force between conductors

Key Concept: Currents flowing in the same direction attract each other, whereas currents flowing in opposite directions repel each other.

🧲 2. Magnetic Field Due to a Long Straight Conductor

Consider a long straight conductor carrying current I₁. The magnetic field at a distance d from the conductor is:

B₁ = μ₀I₁ / 2πd

This magnetic field acts on the second conductor carrying current I₂.

📐 3. Derivation of Force Between Parallel Currents

The second conductor of length L carries current I₂ in the magnetic field produced by the first conductor.

F = BIL sinθ

Since the conductors are parallel:

θ = 90°

Therefore:

F = B₁I₂L

Substituting:

F = (μ₀I₁ / 2πd) I₂L

Hence:

⭐ F = μ₀I₁I₂L / 2πd

Therefore:

⭐ F/L = μ₀I₁I₂ / 2πd

🔬 4. Animated Practical — Parallel Current Conductors

Current I₁ 5.0 A

Current I₂ 5.0 A

Distance d 5.0 m

Current Direction Same Direction Opposite Direction

CONDUCTOR 1

CONDUCTOR 2

F₁

F₂

Force per unit length: 0.000001 N/m
Nature of force: ATTRACTION
Relation: F/L ∝ I₁I₂/d

↔️ 5. Same Direction Currents

When two parallel conductors carry currents in the same direction, the magnetic force between them is attractive.

I₁ ↑     I₂ ↑

✔ Same direction currents → ATTRACT

↔️ 6. Opposite Direction Currents

When two parallel conductors carry currents in opposite directions, the magnetic force between them is repulsive.

I₁ ↑     I₂ ↓

✔ Opposite direction currents → REPEL

⚡ 7. Definition of One Ampere

The classical SI definition of one ampere is based on the force between two parallel current-carrying conductors.

One ampere is that constant current which, when maintained in each of two straight, parallel, infinitely long conductors of negligible circular cross-section placed one metre apart in vacuum, produces between these conductors a force equal to:

⭐ 2 × 10⁻⁷ N per metre length

For I₁ = I₂ = 1 A and d = 1 m:

F/L = μ₀ / 2π

Since:

μ₀ = 4π × 10⁻⁷ N/A²

Therefore:

F/L = 2 × 10⁻⁷ N/m

📚 8. Important Note About Modern SI

Since 20 May 2019, the ampere is defined by fixing the numerical value of the elementary charge e. The parallel-current definition is the classical textbook definition and is still commonly used in school and competitive-exam physics.

📊 9. Dependence of Force

Current I₁

F ∝ I₁

Increasing I₁ increases the force.

Current I₂

F ∝ I₂

Increasing I₂ increases the force.

Distance d

F ∝ 1/d

Increasing separation decreases the force.

Length L

F ∝ L

Force increases with the length of conductor in the magnetic field.

📋 10. Quick Comparison

Current Direction	Force	Nature
Same direction	Towards each other	Attraction
Opposite direction	Away from each other	Repulsion

📝 11. MCQ Practice

1. Two parallel conductors carrying currents in the same direction:

A. Repel each other

B. Attract each other

C. Have no force

D. Rotate around each other

✔ Answer: B

2. The force per unit length between two parallel conductors is:

A. μ₀I₁I₂/2πd

B. μ₀I₁I₂d/2π

C. 2πd/μ₀I₁I₂

D. μ₀d/2πI₁I₂

✔ Answer: A

3. If the distance between two parallel conductors is doubled, the force becomes:

A. Double

B. Half

C. Four times

D. Unchanged

✔ Answer: B

4. If both currents are doubled, the force becomes:

A. 2 times

B. 4 times

C. Half

D. Unchanged

✔ Answer: B

5. The force between parallel current-carrying wires is due to:

A. Electric field only

B. Magnetic field

C. Gravitational field

D. Nuclear force

✔ Answer: B

6. The SI unit of force per unit length is:

A. N

B. N/m

C. N m

D. A/m

✔ Answer: B

7. The magnetic field due to a long straight conductor is:

A. μ₀I/2πr

B. μ₀Ir/2π

C. 2πr/μ₀I

D. μ₀/Ir

✔ Answer: A

8. One ampere classical definition corresponds to force per metre of:

A. 2 × 10⁻⁷ N/m

B. 2 × 10⁷ N/m

C. 9.8 N/m

D. 1 N/m

✔ Answer: A

9. If one current is reversed while the other remains unchanged, the force changes from:

A. Attraction to repulsion

B. Repulsion to attraction

C. Zero to attraction

D. No change

✔ Answer: A

10. The force between two parallel currents is proportional to:

A. I₁ + I₂

B. I₁I₂

C. I₁/I₂

D. 1/I₁I₂

✔ Answer: B

🟣 12. Assertion–Reason — 5 Questions

Options:
A. Both A and R are true, and R is the correct explanation of A.
B. Both A and R are true, but R is not the correct explanation of A.
C. A is true, but R is false.
D. A is false, but R is true.

Assertion: Two parallel conductors carrying currents in the same direction attract each other.

Reason: Each conductor experiences a magnetic force due to the magnetic field produced by the other conductor.

Answer: A

Assertion: The force between two parallel conductors decreases when their separation is increased.

Reason: The magnetic field produced by a long straight conductor is inversely proportional to distance.

Answer: A

Assertion: If both currents are doubled, the force becomes four times.

Reason: The force is directly proportional to the product I₁I₂.

Answer: A

Assertion: Parallel currents flowing in opposite directions attract each other.

Reason: Opposite direction currents produce a repulsive magnetic interaction.

Answer: D

Assertion: The classical definition of one ampere is based on the force between two parallel current-carrying conductors.

Reason: For 1 A current in each conductor separated by 1 m, the force per unit length is 2 × 10⁻⁷ N/m in vacuum.

Answer: A

🟢 13. 2 Marks — 6 Questions

Q1 Write the expression for force per unit length between two parallel current-carrying conductors.

Q2 What happens when two parallel conductors carry currents in the same direction?

Q3 What happens when currents in two parallel conductors flow in opposite directions?

Q4 Define one ampere according to the classical definition.

Q5 Write the magnetic field due to a long straight current-carrying conductor.

Q6 How does force between parallel conductors depend on their separation?

🟡 14. 3 Marks — 6 Questions

Q1 Explain why two parallel conductors carrying currents in the same direction attract each other.

Q2 Explain why two parallel conductors carrying currents in opposite directions repel each other.

Q3 Derive the expression for magnetic field due to a long straight current-carrying conductor.

Q4 Derive the force on a current-carrying conductor placed in a magnetic field.

Q5 Write and explain the factors on which force between two parallel currents depends.

Q6 State the classical definition of one ampere and explain the meaning of 2 × 10⁻⁷ N/m.

🟠 15. 4 Marks — 6 Questions

Q1 Derive the expression for force per unit length between two long, straight, parallel current-carrying conductors.

Q2 Explain attraction and repulsion between parallel current-carrying conductors with suitable diagrams.

Q3 Explain the classical definition of one ampere using the force between parallel conductors.

Q4 Two parallel wires carry currents I₁ and I₂ and are separated by d. Derive the expression for force acting on length L of either wire.

Q5 Explain how the force changes if I₁, I₂ and d are individually doubled.

Q6 Calculate the force per unit length between two wires carrying 2 A and 4 A currents separated by 0.2 m in vacuum.

🔴 16. 5 Marks — 6 Questions

Q1 Derive the expression for force between two parallel current-carrying conductors using the magnetic field of a long straight conductor.

Q2 Explain the experimental basis of the classical definition of one ampere.

Q3 Explain with diagrams why same-direction currents attract and opposite-direction currents repel.

Q4 Two long parallel conductors carry currents of 5 A and 10 A and are separated by 0.5 m. Find the force per metre length between them in vacuum.

Q5 Explain the dependence of force between parallel currents on current, distance and length of the conductors.

Q6 Derive F/L = μ₀I₁I₂/2πd and explain the physical meaning of every term.

🔵 17. 6 Marks — 6 Questions

Q1 State and derive the expression for force between two long straight parallel current-carrying conductors.

Q2 Give a detailed explanation of the classical definition of one ampere. Derive the force per unit length for I₁ = I₂ = 1 A and d = 1 m.

Q3 Explain the magnetic field produced by one current-carrying conductor and use it to derive the force acting on the second conductor.

Q4 Explain attraction and repulsion between parallel current-carrying conductors using the right-hand thumb rule and force equation.

Q5 Derive the force per unit length between two parallel conductors and discuss how the force changes when currents and separation are changed.

Q6 A pair of parallel conductors carry currents I₁ and I₂ at separation d. Derive the complete expression for the force and explain its application in defining the ampere.

🧮 18. Numerical Practice

Example 1

Two parallel wires carry currents:

I₁ = 2 A,   I₂ = 4 A

Their separation is:

d = 0.2 m

F/L = μ₀I₁I₂ / 2πd

F/L = (4π × 10⁻⁷ × 2 × 4)/(2π × 0.2)

⭐ F/L = 8 × 10⁻⁶ N/m

✔ Answer: 8 × 10⁻⁶ N/m

Example 2 — Classical Ampere Definition

For:

I₁ = I₂ = 1 A

d = 1 m

Then:

F/L = μ₀/2π

= (4π × 10⁻⁷)/(2π)

⭐ F/L = 2 × 10⁻⁷ N/m

📚 19. Important Formula Sheet

Magnetic field due to long straight conductor:
B = μ₀I/2πd

Force on current-carrying conductor:
F = BIL sinθ

For parallel conductors:
θ = 90°

Force between parallel currents:
F = μ₀I₁I₂L/2πd

Force per unit length:
⭐ F/L = μ₀I₁I₂/2πd

Classical one ampere:
F/L = 2 × 10⁻⁷ N/m

🚀 20. Quick Revision

🧲 Same Direction

Parallel currents in the same direction attract.

↔️ Opposite Direction

Parallel currents in opposite directions repel.

📐 Main Formula

F/L = μ₀I₁I₂/2πd

⚡ One Ampere

Classically corresponds to 2 × 10⁻⁷ N/m force at 1 m separation.

📏 Distance

F ∝ 1/d

🔌 Currents

F ∝ I₁I₂