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This MCQ module is based on: Biot Savart Amperes Law
TOPIC 15 OF 38
Biot Savart Amperes Law
🎓 Class 12
Physics
CBSE
Theory
Ch 4 – Moving Charges and Magnetism
⏱ ~14 min
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Biot Savart Amperes Law
4.5 Magnetic Field due to a Current Element - Biot-Savart Law
Just as Coulomb's law gives the electric field of a point charge, the Biot-Savart law gives the elementary magnetic field produced by an infinitesimal current element \(I\,d\vec L\) at a point P located at a distance r:
\[d\vec B = \dfrac{\mu_0}{4\pi}\,\dfrac{I\,d\vec L \times \hat r}{r^2}\]
where \(\mu_0 = 4\pi\times10^{-7}\) T m A\(^{-1}\) is the permeability of free space and \(\hat r\) is the unit vector from the element to P.
Magnitude: \(dB = \dfrac{\mu_0}{4\pi}\,\dfrac{I\,dL\,\sin\theta}{r^2}\). Direction: along \(d\vec L \times \hat r\), found by the right-hand rule.
Comparison with Coulomb's law:
- Both fall off as \(1/r^2\).
- Both linear in source strength (q for E, I dL for B).
- E is along \(\hat r\); B is perpendicular to both \(d\vec L\) and \(\hat r\) (cross product).
- \(\mu_0/4\pi = 10^{-7}\) T m/A vs \(1/(4\pi\varepsilon_0) = 9\times10^9\) N m\(^2\)/C\(^2\).
| Aspect | Biot-Savart Law | Ampere's Law |
|---|---|---|
| Form | dB = (mu0/4 pi) I dL x r-hat / r² | integral B . dL = mu0 I_enc |
| Type | Differential (point source) | Integral (closed loop) |
| Best for | Any geometry, especially finite wires & loops | Highly symmetric problems |
| Analogue | Coulomb's law | Gauss's law |
| Universal validity | Yes | Yes (with displacement-current correction) |
4.6 Magnetic Field on the Axis of a Circular Current Loop
Consider a circular loop of radius R carrying current I, with its centre at the origin and axis along x. Apply Biot-Savart to a small element on the loop at distance \(r = \sqrt{R^2+x^2}\) from a point P on the axis. Symmetry kills the components perpendicular to the axis; only the axial components add.
\[B_x = \dfrac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}\]
Special case - centre of the loop (x = 0):
\[B_{centre} = \dfrac{\mu_0 I}{2R}\]
For N closely wound turns, multiply by N: \(B = \mu_0 NI/(2R)\) at the centre.
Worked Example 4.4 - Field at the centre of a coil
Example 4.4 L3 Apply
A circular coil of 200 turns and radius 10 cm carries 0.4 A. Find the magnetic field at its centre.
\(B = \dfrac{\mu_0 NI}{2R} = \dfrac{(4\pi\times10^{-7})(200)(0.4)}{2(0.10)}\)
= \(5.03\times10^{-4}\) T \(\approx 5.0\) gauss.
4.7 Ampere's Circuital Law
For static currents, Ampere proved an extremely useful integral relation - the magnetic analogue of Gauss's law:
\[\oint \vec B \cdot d\vec L = \mu_0 \, I_{enclosed}\]
The line integral of \(\vec B\) around any closed loop equals \(\mu_0\) times the total current threading through any surface bounded by that loop. The closed loop is called an Amperian loop.
4.7.1 Field of a Long Straight Wire
By symmetry B has the same magnitude on a circle of radius r around the wire and is tangent to it. Therefore
\(\oint B\,dL = B(2\pi r) = \mu_0 I \quad\Longrightarrow\quad B = \dfrac{\mu_0 I}{2\pi r}\)
Worked Example 4.5 - Field of a power line
Example 4.5 L3 Apply
A long straight wire carries 35 A. What is the magnitude of B at 20 cm from the wire?
\(B = \dfrac{\mu_0 I}{2\pi r} = \dfrac{(4\pi\times10^{-7})(35)}{2\pi(0.20)} = 3.5\times10^{-5}\) T = 35 microtesla.
About the same as Earth's field - a real concern for sensitive instruments near power lines.
4.8 The Solenoid
A solenoid is a long helical coil. If turns per unit length is n and current is I, applying Ampere's law to a rectangular loop with one side inside and one side far outside gives:
\[B_{inside} = \mu_0 n I, \qquad B_{outside}\approx 0\]
4.8.1 The Toroid
A toroid is a solenoid bent into a closed circle. Apply Ampere's law to a circular Amperian loop of radius r inside the windings (N total turns):
\(B(2\pi r) = \mu_0 N I \quad\Longrightarrow\quad B = \dfrac{\mu_0 N I}{2\pi r}\)
Worked Example 4.6 - Solenoid field
Example 4.6 L3 Apply
A solenoid 0.6 m long has 600 turns and carries 3.0 A. Find B inside.
n = 600/0.6 = 1000 turns/m.
B = mu0 n I = (4 pi x 10-7)(1000)(3.0) = 3.77 x 10-3 T ~ 38 gauss.
Interactive: Magnetic Field along the Axis of a Loop L4 Analyse
Slide the field point along the axis. Watch B fall away from the centre as B = mu0 I R2/(2(R2+x2)3/2).
B = 1.26 x 10-5 T
Activity 4.2 - Compass-and-Wire Demonstration L3 Apply
Place a compass under a long wire connected to a battery (use a series resistor for safety). Switch on the current.
Predict: which way does the needle deflect when current flows from south to north along the wire?
The compass underneath deflects to the east. By the right-hand rule, with thumb pointing N (current direction), fingers curl down on the south side of the wire and up on the north side; below the wire B points east, deflecting the compass needle from north toward east.
Reverse the battery: needle deflects west. This is exactly Oersted's 1820 discovery.
Competency-Based Questions L3-L5
In an MRI scanner, a long superconducting solenoid produces a uniform 1.5 T field along its axis to align nuclear spins. The solenoid is 2.0 m long with 5,000 turns of cooled niobium-titanium wire.
Q1. Which law most efficiently gives the field inside the long solenoid?
(c). Ampere's law exploits the symmetry of an infinite solenoid; Biot-Savart works but is much more laborious.
Q2. (Numerical) What current is needed in the MRI solenoid to produce 1.5 T?
n = 5000/2 = 2500 turns/m. I = B/(mu0 n) = 1.5 / (4 pi x 10-7 x 2500) = 477 A. Only practical with superconductors.
Q3. (Fill in the blank) The magnetic field a perpendicular distance r from a long straight wire is B = ___.
B = mu0 I / (2 pi r). Falls as 1/r.
Q4. (True/False) For a toroid, the magnetic field outside the core is essentially zero.
True. All field lines close inside the doughnut; an Amperian loop drawn outside encloses zero net current.
Q5. (HOT) Two identical loops are placed coaxially distance R apart (one of radius R). Show that the field at the midpoint is more uniform than at the centre of a single loop. (Helmholtz coils)
Add the on-axis field of each loop at x = R/2. Both contributions are equal; calculation shows that dB/dx and d²B/dx² vanish at the midpoint. So B is constant to second order - the basis of Helmholtz coils used to make uniform fields.
Assertion-Reason Questions L4 Analyse
(a) Both A and R true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
A: Inside an ideal long solenoid the magnetic field is uniform.
R: Each turn behaves like a small magnetic dipole and the dipole fields add constructively along the axis.
(a). Stacking many circular currents produces a uniform internal field by superposition; outside they nearly cancel.
A: Ampere's law applies to any closed loop, regardless of geometry.
R: Ampere's law gives a useful direct result only when the field has high symmetry.
(b). Both true but unrelated. The law itself holds always; we can only solve for B easily when we can pull B outside the integral.
A: A circular current loop behaves like a magnetic dipole.
R: At large distances on the axis, B falls as 1/x³, characteristic of a dipole field.
(a). When x >> R, B = mu0 I R²/(2 x³) = mu0 (2 m)/(4 pi x³) with magnetic moment m = I pi R² - exactly a dipole.
Frequently Asked Questions - Biot Savart Amperes Law
What is the main concept covered in Biot Savart Amperes Law?
In NCERT Class 12 Physics Chapter 4 (Moving Charges and Magnetism), "Biot Savart Amperes Law" covers the core principles and equations students need for board exam success. The MyAiSchool lesson explains the topic with definitions, derivations, worked examples, and interactive simulations. Key formulas and dimensional analysis are included to build conceptual depth and problem-solving skills aligned with the CBSE 2025-26 syllabus.
How is Biot Savart Amperes Law useful in real-life applications?
Real-life applications of "Biot Savart Amperes Law" from NCERT Class 12 Physics Chapter 4 include electronics, communication systems, medical imaging, solar energy, semiconductor devices, and modern technology. The MyAiSchool lesson links every concept to a tangible example so students see physics as a problem-solving framework for the physical world, not as abstract formulas.
What are the key formulas in Biot Savart Amperes Law?
Key formulas in "Biot Savart Amperes Law" (NCERT Class 12 Physics Chapter 4 Moving Charges and Magnetism) are derived step-by-step in the MyAiSchool lesson. Students should memorize the final formula AND understand its derivation for full board marks. Each formula is listed with its dimensional formula, SI unit, applicability range, and common pitfalls. The Summary section at the end of each part includes a quick-reference formula card.
How does this part connect to other parts of Chapter 4?
NCERT Class 12 Physics Chapter 4 (Moving Charges and Magnetism) is structured so each part builds on the previous one. "Biot Savart Amperes Law" connects directly to neighbouring parts via shared definitions, units, and methodology. The MyAiSchool lesson cross-references related concepts with internal links so students can navigate the whole chapter as one connected story rather than disconnected fragments.
What types of CBSE board questions come from Biot Savart Amperes Law?
CBSE board questions from "Biot Savart Amperes Law" typically include: (1) 1-mark MCQs on definitions and formulas, (2) 2-mark short-answer derivations or applications, (3) 3-mark numerical problems with units, (4) 5-mark long-answer derivations followed by application. The MyAiSchool lesson tags each Competency-Based Question (CBQ) with Bloom level (L1-L6) so students know how to study for each weight.
How can students use the interactive simulation effectively?
The interactive simulation in the "Biot Savart Amperes Law" lesson allows students to adjust input parameters (sliders or selectors) and see physical quantities update in real time. To use it effectively: (1) try extreme values to understand limiting cases, (2) compare with the analytical formula, (3) check unit consistency, (4) test special configurations from worked examples. The simulation reinforces conceptual intuition that pure formula manipulation cannot.
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