📌 1. Question Description
Biot-Savart Law
The Biot-Savart law gives the magnetic field produced at a point by a
small current element of a current-carrying conductor.
It is the magnetic-field analogue of Coulomb's law and is especially
useful for calculating the magnetic field due to current-carrying wires,
circular loops and other conductors.
dB = (μ₀/4π) × (I dl sinθ / r²)
For a circular current-carrying loop, the contributions of all current
elements are added using the principle of superposition.
At the centre of a circular loop of radius R carrying current I:
B = μ₀I / 2R
For a coil having N turns:
B = μ₀NI / 2R
⚡ 2. Principle of Biot-Savart Law
According to Biot-Savart law, the magnetic field dB produced at a point
by a current element I dl is:
dB ∝ I
dB ∝ dl
dB ∝ sinθ
dB ∝ 1/r²
Combining all these relations:
dB = (μ₀/4π) (I dl sinθ/r²)
where:
- I = current in the conductor
- dl = small current element
- r = distance of observation point from current element
- θ = angle between dl and r
- μ₀ = permeability of free space
🧭 3. Direction of Magnetic Field
The direction of magnetic field due to a current element is perpendicular
to the plane containing the current element and the position vector.
Its direction is determined by the right-hand thumb rule.
👉 Curl the fingers of your right hand in the direction of current in the
circular loop. Your thumb gives the direction of magnetic field at the
centre of the loop.
🔬 4. Animated Circular Loop Practical
Current:
3.00 A
Radius:
20.00 cm
Turns:
5
Magnetic Field at Centre:
4.71 × 10⁻⁵ T
MAGNETIC FIELD GENERATED ✔
📐 5. Magnetic Field at the Centre of a Circular Loop
Consider a circular loop of radius R carrying current I.
For every small element dl of the circular loop:
θ = 90°
Therefore:
sin90° = 1
Using Biot-Savart law:
dB = (μ₀/4π)(I dl/R²)
Since every current element produces magnetic field in the same direction
at the centre:
B = ∫dB
B = (μ₀I/4πR²) ∫dl
For a complete circular loop:
∫dl = 2πR
Therefore:
B = (μ₀I/4πR²)(2πR)
⭐ B = μ₀I/2R
🔄 6. Magnetic Field Due to a Coil of N Turns
If the circular coil has N identical turns, the magnetic field due to
each turn adds in the same direction.
B = N × μ₀I/2R
Therefore:
⭐ B = μ₀NI/2R
Thus magnetic field increases with:
- Current I
- Number of turns N
and decreases with:
📍 7. Magnetic Field on the Axis of a Circular Loop
At a point P on the axis of a circular loop, at distance x from its centre,
the magnetic field is:
B = μ₀NIR² / [2(R² + x²)3/2]
At the centre, x = 0:
B = μ₀NI/2R
Important:
The magnetic field is maximum at the centre of the circular loop and
decreases as we move away from the centre along its axis.
📊 8. Dependence of Magnetic Field
Current
B ∝ I
If current is doubled, magnetic field becomes double.
Number of Turns
B ∝ N
If the number of turns is doubled, magnetic field becomes double.
Radius
B ∝ 1/R
For the same current and turns, a smaller radius produces a larger field
at the centre.
🖐️ 9. Right-Hand Thumb Rule
Hold the circular conductor in your right hand such that your curled
fingers point in the direction of current.
Your extended thumb gives the direction of magnetic field at the centre
of the loop.
Clockwise current → Magnetic field into the plane.
Anticlockwise current → Magnetic field out of the plane.
📝 10. MCQ Practice
1. Biot-Savart law is used to determine:
A. Electric field due to charge
B. Magnetic field due to current element
C. Electric potential only
D. Resistance
✔ Answer: B
2. The magnetic field at the centre of a circular loop is:
A. μ₀I/R
B. μ₀I/2R
C. μ₀IR
D. μ₀/IR
✔ Answer: B
3. Magnetic field at the centre of a coil of N turns is:
A. μ₀I/2R
B. μ₀NI/2R
C. μ₀IR/2N
D. μ₀NR/2I
✔ Answer: B
4. The magnetic field at the centre of a circular loop is directly
proportional to:
A. R
B. 1/I
C. I
D. R²
✔ Answer: C
5. If the radius of a circular loop is doubled, keeping current constant,
the field at the centre becomes:
A. Double
B. Half
C. Four times
D. Unchanged
✔ Answer: B
6. In Biot-Savart law, magnetic field varies with distance as:
A. r
B. r²
C. 1/r²
D. 1/r
✔ Answer: C
7. For a current element at the centre of a circular loop:
A. θ = 0°
B. θ = 45°
C. θ = 90°
D. θ = 180°
✔ Answer: C
8. The direction of magnetic field at the centre of a circular loop is
found using:
A. Fleming's left-hand rule
B. Right-hand thumb rule
C. Lenz's law
D. Kirchhoff's law
✔ Answer: B
9. If both current and number of turns are doubled, magnetic field
at the centre becomes:
A. 2 times
B. 3 times
C. 4 times
D. Half
✔ Answer: C
10. The SI unit of magnetic field is:
A. Weber
B. Tesla
C. Henry
D. Ampere
✔ Answer: B
🟣 11. Assertion–Reason Practice — 5 Questions
Choose the correct option:
A. Both Assertion and Reason are true, and Reason is the correct
explanation of Assertion.
B. Both Assertion and Reason are true, but Reason is not the correct
explanation.
C. Assertion is true, but Reason is false.
D. Assertion is false, but Reason is true.
Assertion (A):
Magnetic field at the centre of a circular loop is directly proportional
to current.
Reason (R):
According to Biot-Savart law, dB is directly proportional to current I.
Answer: A
Assertion (A):
Magnetic field at the centre of a circular loop decreases when its radius
is increased.
Reason (R):
For a circular loop, B = μ₀NI/2R.
Answer: A
Assertion (A):
The magnetic field at the centre of a circular loop is maximum on its axis.
Reason (R):
The magnetic field decreases as the distance from the centre increases.
Answer: D
Assertion (A):
The magnetic field due to every current element of a circular loop at its
centre has the same direction.
Reason (R):
The angle between dl and r is 90° for every element at the centre.
Answer: B
Assertion (A):
Increasing the number of turns of a circular coil increases the magnetic
field at its centre.
Reason (R):
The magnetic fields produced by individual turns add together.
Answer: A
🟢 12. 2 Marks — 6 Questions
Q1
State Biot-Savart law and write its mathematical expression.
Q2
Define a current element.
Q3
Write the expression for magnetic field at the centre of a circular loop.
Q4
How does the magnetic field at the centre of a circular loop depend on
current and radius?
Q5
State the right-hand thumb rule.
Q6
Write the expression for magnetic field at the centre of a circular coil
having N turns.
🟡 13. 3 Marks — 6 Questions
Q1
Explain Biot-Savart law and define all the quantities appearing in its
mathematical expression.
Q2
Derive the magnetic field at the centre of a circular current-carrying
loop.
Q3
Explain the direction of magnetic field at the centre of a circular loop.
Q4
How does magnetic field change if the current and radius of a circular
loop are changed?
Q5
Why is the magnetic field due to all current elements in a circular loop
in the same direction at the centre?
Q6
Explain why increasing the number of turns increases the magnetic field.
🟠 14. 4 Marks — 6 Questions
Q1
State and explain Biot-Savart law with its vector form.
Q2
Derive the expression B = μ₀I/2R for a circular loop.
Q3
Derive the expression for magnetic field at the centre of a coil of N turns.
Q4
Explain the factors affecting the magnetic field at the centre of a
circular current-carrying loop.
Q5
A circular loop of radius 20 cm carries a current of 3 A. Calculate the
magnetic field at its centre.
Q6
Explain the right-hand thumb rule and apply it to a circular current loop.
🔴 15. 5 Marks — 6 Questions
Q1
State Biot-Savart law and derive the magnetic field at the centre of a
circular current-carrying loop.
Q2
Derive the expression for magnetic field at the centre of a circular coil
of N turns.
Q3
Explain the dependence of magnetic field on current, radius and number
of turns of a circular loop.
Q4
Explain the direction of magnetic field due to a current-carrying circular
loop using the right-hand thumb rule.
Q5
A coil of 10 turns and radius 10 cm carries a current of 2 A. Calculate
the magnetic field at its centre.
Q6
Explain Biot-Savart law and discuss its application to a circular
current-carrying loop.
🔵 16. 6 Marks — 6 Questions
Q1
State Biot-Savart law in vector form and derive the magnetic field at the
centre of a circular current-carrying loop.
Q2
Using Biot-Savart law, derive the expression for magnetic field at a point
on the axis of a circular current-carrying coil.
Q3
Derive the magnetic field at the centre of a circular coil of N turns and
explain the effect of current, radius and number of turns.
Q4
Explain the complete application of Biot-Savart law to a circular loop
with a suitable labelled diagram and derivation.
Q5
A circular coil of 20 turns and radius 15 cm carries a current of 4 A.
Calculate the magnetic field at its centre. Also state its direction.
Q6
Compare the magnetic field at the centre and at a point on the axis of
a circular current-carrying loop. Derive the relevant expressions.
🧮 17. Numerical Practice
Example 1
A circular loop of radius 20 cm carries a current of 3 A. Find the
magnetic field at its centre.
B = μ₀I/2R
R = 20 cm = 0.20 m
B = (4π × 10⁻⁷ × 3)/(2 × 0.20)
B ≈ 9.42 × 10⁻⁶ T
✔ Answer: B ≈ 9.42 × 10⁻⁶ T
Example 2
A circular coil of 10 turns, radius 10 cm, carries a current of 2 A.
Find the magnetic field at its centre.
B = μ₀NI/2R
B =
(4π × 10⁻⁷ × 10 × 2)/(2 × 0.10)
B ≈ 1.256 × 10⁻⁴ T
✔ Answer: B ≈ 1.26 × 10⁻⁴ T
📚 18. Important Formula Sheet
Biot-Savart Law:
dB = (μ₀/4π)(I dl sinθ/r²)
Vector Form:
d𝐁 = (μ₀/4π) [I(d𝐥 × r̂)/r²]
Single Circular Loop — Centre:
B = μ₀I/2R
N-Turn Circular Coil:
B = μ₀NI/2R
On Axis:
B = μ₀NIR²/[2(R²+x²)3/2]
🚀 19. Quick Revision
🔹 Biot-Savart Law
Gives magnetic field due to a current element.
🔹 Centre of Loop
B = μ₀NI/2R
🔹 Direction
Use right-hand thumb rule.
