Magnetic field on the axis of a circular current loop
Derive and apply the formula for the magnetic field strength at a point on the axis of a circular current-carrying loop using Biot–Savart Law.
What is Magnetic field on the axis of a circular current loop?
A fundamental law in electromagnetism that describes the magnetic field generated by a steady electric current.
Key formula / rule: Magnetic field element (Biot-Savart Law)
Key points
- Understand the application of Biot-Savart Law to calculate magnetic fields.
- Derive the formula for the magnetic field on the axis of a circular current loop.
- Apply the derived formula to solve problems involving circular current loops.
- Determine the direction of the magnetic field using the right-hand rule.
Common exam trap
Incorrectly applying the cross product in Biot-Savart Law.
Definitions
- Term
Biot-Savart Law
- Meaning
A fundamental law in electromagnetism that describes the magnetic field generated by a steady electric current.
- Term
Circular Current Loop
- Meaning
A closed loop of wire in the shape of a circle carrying an electric current.
- Term
Axis of a Loop
- Meaning
An imaginary line passing through the center of the loop and perpendicular to its plane.
- Term
Permeability of Free Space (μ₀)
- Meaning
A physical constant representing the measure of the ability of a vacuum to support the presence of a magnetic field.
Learning objectives
Understand the application of Biot-Savart Law to calculate magnetic fields.
Derive the formula for the magnetic field on the axis of a circular current loop.
Apply the derived formula to solve problems involving circular current loops.
Determine the direction of the magnetic field using the right-hand rule.
Formulae
- Name
Magnetic field element (Biot-Savart Law)
- Note
μ₀ is the permeability of free space, I is current, dl is the current element vector, r̂ is the unit vector from dl to the point, r is the distance.
- Expression
dB = (μ₀/4π) * (Idl × r̂)/r²
- Name
Distance from current element to point on axis
- Note
R is the radius of the loop, x is the distance from the center of the loop to the point on the axis.
- Expression
r = √(R² + x²)
- Name
Magnetic field on the axis of a circular loop
- Note
I is the current, R is the radius of the loop, x is the distance from the center along the axis.
- Expression
B = (μ₀IR²)/(2(R² + x²)^(3/2))
- Name
Magnetic field at the center of a circular loop
- Note
This is a special case of the axial field formula when x = 0.
- Expression
B = μ₀I/(2R)
Prerequisites
Biot-Savart Law
Vector cross product
Basic calculus (integration)
Right-hand rule for magnetic fields
Common mistakes
Incorrectly applying the cross product in Biot-Savart Law.
Forgetting to resolve the magnetic field components.
Errors in integration or setting up the limits of integration.
Confusing the distance 'r' from the element to the point with the radius 'R' of the loop.
Incorrectly calculating cos θ or sin θ.
Not considering the cancellation of perpendicular components.
Keywords
Biot-Savart Law
Circular Current Loop
Magnetic Field
Axis
Ampere's Law
Electromagnetism
Current Element
Permeability of Free Space
Practice preview
For a circular coil of radius R carrying a current I, what is the magnetic field at the center of the coil?…
easy
A circular coil of radius R is placed on the xy-plane with its center at the origin. A current I flows through the coil. What is the direction of the magnetic field at a point on the positive z-axis?…
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Which of the following statements is INCORRECT regarding the magnetic field on the axis of a circular current loop?…
medium