🧲 Chapter 56: Application of Biot–Savart Law to a Current-Carrying Circular Loop (Class XII)
🔷 1. Introduction
One of the most important applications of the Biot–Savart Law is the calculation of the magnetic field produced by a current-carrying circular loop. Since every small current element of the loop produces a magnetic field, the total magnetic field at the center or any point on the axis of the loop is obtained by adding the contributions of all current elements.
This application forms the basis for understanding the working of electromagnets, galvanometers, electric motors, solenoids, and circular coils.
🔷 2. Current-Carrying Circular Loop
Ethan: Professor, what is meant by a current-carrying circular loop?
Professor: It is a circular conducting wire through which electric current flows. Every small element of the wire produces a magnetic field, and together they produce a strong magnetic field at the center of the loop.
Academic Definition
A current-carrying circular loop is a circular conductor carrying electric current that produces a magnetic field around it, with the magnetic field being strongest at its center.
🔷 3. Applying the Biot–Savart Law
Ethan: Professor, how is the Biot–Savart Law applied to a circular loop?
Professor: Consider a very small element dl of the circular loop carrying current I. According to the Biot–Savart Law, this small element produces a small magnetic field dB at the center of the loop.
For a circular loop, the current element is always perpendicular to the radius joining the element to the center.
θ = 90°
Since,
sin 90° = 1
the Biot–Savart equation becomes much simpler.
🔷 4. Magnetic Field Due to a Small Current Element
Ethan: Professor, what is the magnetic field due to one small current element?
Professor:
dB = (μ₀/4π) × (I dl/R²)
where,
- I = Current in the loop.
- dl = Small length element.
- R = Radius of the circular loop.
- μ₀ = Permeability of free space.
🔷 5. Total Magnetic Field at the Centre
Ethan: Professor, how do we obtain the magnetic field due to the entire loop?
Professor: We add the magnetic fields produced by all the current elements around the loop. Since all the magnetic field vectors at the center are in the same direction, they add directly.
The total length of the circular loop is
2πR
Substituting this into the Biot–Savart expression gives the magnetic field at the center of the loop.
B = μ₀I / 2R
This is the magnetic field at the center of a circular loop carrying current I.
🔷 6. Magnetic Field for N Turns
Ethan: Professor, what happens if the loop has several turns?
Professor: Every turn produces the same magnetic field. Therefore, the total magnetic field becomes N times larger.
B = μ₀NI / 2R
where,
- N = Number of turns.
- I = Current.
- R = Radius of the loop.
🔷 7. Direction of Magnetic Field
Ethan: Professor, how do we determine the direction of the magnetic field?
Professor: The direction is obtained by the Right-Hand Thumb Rule.
- Curl the fingers of the right hand in the direction of current.
- The extended thumb points in the direction of the magnetic field at the center of the loop.
🔷 8. Factors Affecting Magnetic Field
Ethan: Professor, on which factors does the magnetic field depend?
Professor:
- It is directly proportional to the current (I).
- It is directly proportional to the number of turns (N).
- It is inversely proportional to the radius of the loop (R).
B ∝ NI
B ∝ 1/R
🔷 9. Special Cases
| Condition | Effect on Magnetic Field |
|---|---|
| Current increases | Magnetic field increases. |
| Radius increases | Magnetic field decreases. |
| Number of turns increases | Magnetic field increases proportionally. |
| No current | Magnetic field becomes zero. |
🔷 10. Applications
- Electromagnets.
- Moving-coil galvanometers.
- Electric motors.
- Loudspeakers.
- Magnetic sensors.
- Helmholtz coils.
- Magnetic resonance instruments.
📦 11. Important Results (Must Remember)
- Biot–Savart Law is used to calculate the magnetic field due to a circular loop.
- For a circular loop, the angle between dl and the radius is 90°.
- The magnetic field at the center of a single-turn loop is B = μ₀I/2R.
- The magnetic field at the center of an N-turn loop is B = μ₀NI/2R.
- The magnetic field is directly proportional to the current.
- The magnetic field is directly proportional to the number of turns.
- The magnetic field is inversely proportional to the radius.
- The direction of the magnetic field is given by the Right-Hand Thumb Rule.
🧠 12. Conceptual Questions
🔹 Q1
Ethan: Why is θ equal to 90° for a circular loop?
Professor: Because the current element is tangential to the loop, while the radius is directed toward the center. A tangent is always perpendicular to the radius at the point of contact.
🔹 Q2
Ethan: What is the magnetic field at the center of a single circular loop?
Professor: B = μ₀I/2R.
🔹 Q3
Ethan: How does increasing the radius affect the magnetic field?
Professor: The magnetic field decreases because it is inversely proportional to the radius.
🔹 Q4
Ethan: How does increasing the number of turns affect the magnetic field?
Professor: The magnetic field increases directly in proportion to the number of turns.
🔹 Q5
Ethan: Which rule determines the direction of the magnetic field?
Professor: The Right-Hand Thumb Rule.
🔷 13. Summary
The Biot–Savart Law is used to determine the magnetic field produced by a current-carrying circular loop. Since every current element contributes equally to the magnetic field at the center and all contributions are in the same direction, the resultant magnetic field is B = μ₀I/2R for a single-turn loop and B = μ₀NI/2R for an N-turn loop. The magnetic field increases with current and the number of turns but decreases with increasing radius. The direction of the magnetic field is determined using the Right-Hand Thumb Rule.
✨ End of Topic: Application of Biot–Savart Law to a Current-Carrying Circular Loop ✨
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