CUET UG Physics: Magnetism - Biot-Savart Law, Ampere's Law, Solenoid and Toroid

Must-Know

• Biot-Savart Law states that the magnetic field ( d\vec{B} ) due to a current element ( I d\vec{l} ) at a point with position vector ( \vec{r} ) is ( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} ), where ( \mu_0 = 4\pi \times 10^{-7}~\text{T m A}^{-1} ). Example: magnetic field at distance ( R ) from infinite straight wire is ( \frac{\mu_0 I}{2\pi R} ).
• The direction of ( d\vec{B} ) is given by the right-hand thumb rule for cross product ( d\vec{l} \times \hat{r} ).
• Magnetic field due to a circular arc of wire subtending angle ( \theta ) (in radians) at center is ( B = \frac{\mu_0 I \theta}{4\pi R} ); for full circle (( \theta = 2\pi )), ( B = \frac{\mu_0 I}{2R} ).
• Ampere’s Circuital Law: ( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} ), where ( I_{\text{enc}} ) is the net current enclosed by the Amperian loop.
• For an infinitely long straight conductor, using Ampere’s law with circular path gives ( B = \frac{\mu_0 I}{2\pi r} ).
• Magnetic field inside a long solenoid is uniform and axial: ( B = \mu_0 n I ), where ( n = ) number of turns per unit length.
• A solenoid behaves like a bar magnet; one end acts as N-pole and other as S-pole depending on current direction (use right-hand grip rule).
• The field outside an ideal solenoid is nearly zero.
• For a toroid with ( N ) total turns and average radius ( r ), magnetic field inside is ( B = \frac{\mu_0 N I}{2\pi r} ); it varies inversely with ( r ).
• Magnetic field in a toroid is zero outside the core and in the central hole.
• The permeability of free space ( \mu_0 = 4\pi \times 10^{-7}~\text{T m A}^{-1} ).
• Biot-Savart Law is analogous to Coulomb’s Law in electrostatics but for magnetic fields due to current elements.
• Superposition principle applies for magnetic fields: total ( \vec{B} ) is vector sum of fields from all current elements.
• In a solenoid, if current increases, magnetic field strength increases proportionally (since ( B \propto I )).
• Toroidal coils are used in transformers and tokamaks because they confine magnetic field entirely within the core.
• The magnetic field due to a straight wire decreases as ( 1/r ), while electric field due to line charge also decreases as ( 1/r ).
• Ampere’s Law is valid only for steady currents (magnetostatics).
• A solenoid with iron core becomes an electromagnet with greatly enhanced magnetic field due to high permeability of iron.
• The magnetic field at the center of a circular loop of radius ( R ) carrying current ( I ) is ( B = \frac{\mu_0 I}{2R} ).
• For a toroid, if ( N = 1000 ), ( I = 2~\text{A} ), and ( r = 0.5~\text{m} ), then ( B = \frac{(4\pi \times 10^{-7}) \times 1000 \times 2}{2\pi \times 0.5} = 8 \times 10^{-4}~\text{T} ) (verify from NCERT).

Difficulty Level

Intermediate — requires conceptual clarity between Biot-Savart and Ampere’s Law applications, and visualization of field patterns in solenoid and toroid.

Common CUET Traps

• Trap: Assuming Ampere’s Law can be applied easily to any shape like a square loop to find B. Avoid: Ampere’s Law is useful only when symmetry allows ( B ) to be constant along the path (e.g., infinite straight wire, solenoid, toroid).
• Trap: Thinking magnetic field outside a solenoid is strong. Avoid: For an ideal long solenoid, the external field is negligible; most field lines are confined inside.
• Trap: Using ( B = \mu_0 n I ) for a short solenoid. Avoid: This formula applies only to long solenoids where length-diameter and near the center.

Practice MCQs

1. Question: What is the magnetic field at a distance of 0.1 m from a long straight wire carrying a current of 5 A?
   A) ( 10^{-5}~\text{T} )
   B) ( 2 \times 10^{-5}~\text{T} )
   C) ( 10^{-6}~\text{T} )
   D) ( 5 \times 10^{-5}~\text{T} )
   Answer: A
   Explanation: ( B = \frac{\mu_0 I}{2\pi r} = \frac{4\pi \times 10^{-7} \times 5}{2\pi \times 0.1} = 10^{-5}~\text{T} ).
   Why others fail: Option B might be chosen by misplacing decimal or doubling the value incorrectly.

2. Question: Which law gives the magnetic field due to a current-carrying circular loop at its center most directly?
   A) Gauss’s Law
   B) Ampere’s Law
   C) Biot-Savart Law
   D) Faraday’s Law
   Answer: C
   Explanation: Biot-Savart Law is used to derive the field at the center of a circular loop.
   Why others fail: Ampere’s Law lacks symmetry here, so it cannot be applied directly.

3. Question: The magnetic field inside a long solenoid depends on:
   A) Diameter of the solenoid
   B) Material of the wire
   C) Current and turns per unit length
   D) Total number of turns only
   Answer: C
   Explanation: ( B = \mu_0 n I ), independent of diameter or total length.
   Why others fail: Students confuse total turns with turns per unit length (n).

4. Question: In a toroid, the magnetic field is zero:
   A) At the outer edge
   B) Inside the central hole and outside the toroid
   C) Only outside the toroid
   D) Only at the center of each turn
   Answer: B
   Explanation: Magnetic field lines are confined within the toroidal core; zero in hole and external region.
   Why others fail: Option C ignores the field is also zero in the central hole.

5. Question: An Amperian loop encloses three wires: two carrying current ( I ) out of the plane and one carrying ( 2I ) into the plane. What is ( \oint \vec{B} \cdot d\vec{l} )?
   A) 0
   B) ( \mu_0 I )
   C) ( 2\mu_0 I )
   D) ( \mu_0 (3I) )
   Answer: A
   Explanation: Net enclosed current = ( I + I - 2I = 0 ), so ( \oint \vec{B} \cdot d\vec{l} = \mu_0 \times 0 = 0 ).
   Why others fail: Option B is tempting if one assumes net current is ( I ) without proper sign convention.

Last?Minute Revision

• ( B = \frac{\mu_0 I}{2\pi r} ): infinite straight wire — use right-hand rule for direction.
• Biot-Savart: ( d\vec{B} \propto I d\vec{l} \times \hat{r}/r^2 ); direction from cross product.
• Magnetic field at center of circular loop: ( B = \frac{\mu_0 I}{2R} ).
• Ampere’s Law: ( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} ) — only for steady currents.
• Solenoid: ( B = \mu_0 n I ) — uniform and axial inside.
• Toroid: ( B = \frac{\mu_0 N I}{2\pi r} ) — depends on radial distance.
• Field outside ideal solenoid-0.
• Field outside toroid = 0 and in central hole = 0.
• ( \mu_0 = 4\pi \times 10^{-7}~\text{T m A}^{-1} ) — must remember value.
• Right-hand grip rule: for solenoid, fingers show current, thumb shows N-pole.
• Toroid has no free ends — unlike solenoid.
• Ampere’s Law fails for finite wires due to lack of symmetry.
• Biot-Savart is fundamental; Ampere’s is derived under symmetry.
• Solenoid field independent of cross-sectional area.
• In toroid, B-1/r — not uniform across cross-section.
• Use Ampere’s Law for solenoid with rectangular loop crossing the boundary.
• For circular symmetry (wire), use Ampere’s Law; for bent wires, use Biot-Savart.
• Net current in Ampere’s Law uses algebraic sum with sign based on direction.
• Remember: ( \frac{\mu_0}{4\pi} = 10^{-7}~\text{T m A}^{-1} ) — useful in calculations.
• Mnemonic: “Solenoid = Straight field inside”; “Toroid = Totally trapped field”.