Magnetic Dipole Moment: Dipole moment of a revolving electron and a current loop

📌 1. Question Description

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnetic dipole.

For a current-carrying loop, magnetic dipole moment is defined as:

μ = IA

For a coil having N turns:

⭐ μ = NIA

Its direction is perpendicular to the plane of the loop and is given by the right-hand thumb rule.

🔄 2. Magnetic Dipole Moment of a Current Loop

Consider a current I flowing through a circular loop of area A. The magnetic dipole moment of the loop is:

μ = IA

If the loop has N turns:

⭐ μ = NIA

For a circular loop of radius r:

A = πr²

Therefore:

⭐ μ = NIπr²

SI unit of magnetic dipole moment = A m²

🧭 3. Direction of Magnetic Dipole Moment

The direction of magnetic dipole moment is perpendicular to the plane of the current loop.

It is determined by the Right-Hand Thumb Rule:

• Curl the fingers of the right hand in the direction of current.
• The thumb points in the direction of magnetic dipole moment.

μ⃗ = IA n̂

⚛️ 4. Magnetic Dipole Moment of a Revolving Electron

An electron revolving around the nucleus constitutes a current loop. Therefore, it produces a magnetic dipole moment.

Let an electron of charge magnitude e revolve with angular velocity ω in a circular orbit of radius r.

Time period of revolution:

T = 2π/ω

Current due to revolving electron:

I = e/T

Therefore:

I = eω/2π

Area of the orbit:

A = πr²

Hence:

μ = IA

⭐ μ = eωr²/2

📐 5. Relation Between Magnetic Dipole Moment and Angular Momentum

For an electron revolving in a circular orbit, its angular momentum is:

L = mₑvr

The magnetic dipole moment is:

μ = evr/2

Therefore:

⭐ μ/L = e/2mₑ

Hence:

⭐ μ = (e/2mₑ)L

For an electron, because its charge is negative, the magnetic dipole moment is opposite in direction to its orbital angular momentum.

μ⃗ = −(e/2mₑ)L⃗

⚛️ 6. Bohr Magneton

The natural unit of atomic magnetic moment is called the Bohr magneton.

⭐ μ_B = eh/4πmₑ

Its approximate value is:

μ_B ≈ 9.27 × 10⁻²⁴ A m²

🔬 7. Animated Practical — Current Loop

Current I 5 A

Number of Turns N 5

Radius r 5 cm

μ →

Current I →

Magnetic Dipole Moment: 0.0393 A m²

Formula: μ = NIπr²

Observation: Increasing current increases magnetic dipole moment.

📊 8. Current Loop vs Revolving Electron

Current Loop	Revolving Electron
μ = IA	μ = evr/2
For N turns: μ = NIA	Electron forms microscopic current loop
Direction by right-hand rule	μ is opposite to orbital angular momentum
SI unit: A m²	Often expressed in Bohr magneton

⭐ 9. Important Points

1️⃣ Formula

μ = IA

2️⃣ N Turns

μ = NIA

3️⃣ Circular Loop

μ = NIπr²

4️⃣ Direction

Perpendicular to plane of loop.

5️⃣ Electron

μ = evr/2

6️⃣ Angular Momentum

μ = (e/2mₑ)L

📝 10. MCQ Practice — 10 Questions

1. Magnetic dipole moment of a current loop is:

A. I/A

B. IA

C. A/I

D. I + A

✔ Answer: B

2. Magnetic dipole moment of N-turn coil is:

A. IA/N

B. NIA

C. NI/A

D. A/NI

✔ Answer: B

3. SI unit of magnetic dipole moment is:

A. Tesla

B. Weber

C. A m²

D. N/C

✔ Answer: C

4. Direction of magnetic dipole moment of a current loop is:

A. Along the plane of loop

B. Perpendicular to the plane of loop

C. Always downward

D. Always horizontal

✔ Answer: B

5. For a circular coil of radius r, magnetic moment is:

A. NIπr

B. NIπr²

C. NI/r²

D. NIr

✔ Answer: B

6. A revolving electron behaves like:

A. Electric capacitor

B. Current loop

C. Resistor

D. Transformer

✔ Answer: B

7. Magnetic moment of an electron is related to its angular momentum by:

A. μ = 2mL/e

B. μ = eL/2m

C. μ = eL/m

D. μ = mL/e

✔ Answer: B

8. For an electron, magnetic moment and orbital angular momentum are:

A. Parallel

B. Antiparallel

C. Always perpendicular

D. Unrelated

✔ Answer: B

9. The natural unit of atomic magnetic moment is:

A. Tesla

B. Bohr magneton

C. Ampere

D. Weber

✔ Answer: B

10. Bohr magneton is approximately:

A. 9.27 × 10⁻²⁴ A m²

B. 9.27 × 10²⁴ A m²

C. 1.6 × 10⁻¹⁹ A m²

D. 6.67 × 10⁻¹¹ A m²

✔ Answer: A

🟣 11. 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: The magnetic dipole moment of a current loop is proportional to its area.

Reason: μ = IA.

Answer: A

Assertion: Magnetic dipole moment increases when the number of turns of a coil is increased.

Reason: For an N-turn coil, μ = NIA.

Answer: A

Assertion: The magnetic dipole moment of an electron is opposite to its orbital angular momentum.

Reason: The electron has negative charge.

Answer: A

Assertion: Magnetic dipole moment of a circular loop is proportional to its radius.

Reason: Area of circular loop is πr².

Answer: D

Assertion: Bohr magneton is used as a unit of atomic magnetic moment.

Reason: Electrons possess orbital magnetic moments.

Answer: A

🟢 12. 2 Marks — 6 Questions

Q1 Define magnetic dipole moment of a current loop.

Q2 Write the SI unit and dimensional formula of magnetic dipole moment.

Q3 Write the expression for magnetic dipole moment of an N-turn circular coil.

Q4 State the direction of magnetic dipole moment of a current loop.

Q5 What is a Bohr magneton?

Q6 Write the relation between orbital magnetic moment and orbital angular momentum of an electron.

🟡 13. 3 Marks — 6 Questions

Q1 Derive the expression μ = IA for a current loop.

Q2 Derive the magnetic dipole moment of an electron revolving in a circular orbit.

Q3 Derive the relation μ/L = e/2mₑ for a revolving electron.

Q4 Explain the direction of magnetic dipole moment using the right-hand thumb rule.

Q5 Explain why a revolving electron produces a magnetic dipole moment.

Q6 Write the expression for Bohr magneton and state its value.

🟠 14. 4 Marks — 6 Questions

Q1 Derive the expression for magnetic dipole moment of a circular current loop.

Q2 Derive the expression for magnetic moment of an electron revolving in a circular orbit.

Q3 Establish the relation between magnetic dipole moment and angular momentum of an electron.

Q4 Explain the direction and SI unit of magnetic dipole moment.

Q5 Explain the concept of Bohr magneton and its significance.

Q6 Compare the magnetic dipole moment of a current loop and a revolving electron.

🔴 15. 5 Marks — 6 Questions

Q1 Derive the expression for magnetic dipole moment of an N-turn circular current-carrying loop.

Q2 Derive the expression for the magnetic dipole moment of a revolving electron and explain its direction.

Q3 Derive the relation between orbital magnetic moment and orbital angular momentum of an electron.

Q4 Explain magnetic dipole moment, its direction, SI unit and dimensional formula.

Q5 What is Bohr magneton? Derive its expression and state its approximate value.

Q6 Explain how a current loop behaves as a magnetic dipole and derive its magnetic moment.

🔵 16. 6 Marks — 6 Questions

Q1 Derive the expression for magnetic dipole moment of a current-carrying circular loop and explain its direction using the right-hand thumb rule.

Q2 An electron revolves around the nucleus in a circular orbit. Derive the expression for its magnetic dipole moment.

Q3 Derive the relation between magnetic dipole moment and angular momentum of a revolving electron. Explain why their directions are opposite.

Q4 Define Bohr magneton. Derive its expression from the orbital magnetic moment of an electron.

Q5 Explain in detail the analogy between a current loop and a magnetic dipole. Obtain the expression for its magnetic dipole moment.

Q6 A circular coil has N turns, radius r and carries current I. Derive the expression for its magnetic dipole moment and discuss the effect of changing N, r and I.

🧮 17. Numerical Practice

Numerical 1

A circular coil of 10 turns and radius 10 cm carries a current of 2 A. Find its magnetic dipole moment.

μ = NIπr²

μ = 10 × 2 × π × (0.1)²

⭐ μ ≈ 0.628 A m²

✔ Answer: 0.628 A m²

Numerical 2

A current of 5 A flows through a circular loop of area 0.02 m². Find its magnetic dipole moment.

μ = IA

μ = 5 × 0.02

⭐ μ = 0.10 A m²

✔ Answer: 0.10 A m²

🚀 18. Quick Revision

🔄 Current Loop

μ = IA

🌀 N-Turn Coil

μ = NIA

⭕ Circular Coil

μ = NIπr²

⚛️ Revolving Electron

μ = evr/2

📐 Angular Momentum

μ = eL/2mₑ

⭐ Bohr Magneton

μ_B = eh/4πmₑ

📚 19. Formula Sheet

Current loop:
μ = IA

N-turn coil:
μ = NIA

Circular coil:
μ = NIπr²

Revolving electron:
μ = evr/2

Angular momentum:
L = mₑvr

Magnetic moment/angular momentum:
μ/L = e/2mₑ

Bohr magneton:
μ_B = eh/4πmₑ

SI Unit:
A m²