Biot–Savart and Ampere's Circuital Law
Field of a straight wire, circular loop, solenoid and toroid. (Physics › Magnetic Effects of Current and Magnetism, NEET UG syllabus.)
What is Biot–Savart and Ampere's Circuital Law?
A law that describes the magnetic field generated by an electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.
Key formula / rule: Biot–Savart Law (Differential Form)
Key points
- State and apply Biot–Savart Law to calculate the magnetic field due to a current element.
- Derive and apply the magnetic field formulas for a straight current-carrying wire (finite and infinite).
- Derive and apply the magnetic field formulas for a circular current loop (at center and on axis).
- State and apply Ampere's Circuital Law to calculate magnetic fields for symmetric current distributions.
Common exam trap
Incorrectly applying the Right-Hand Thumb Rule for direction of magnetic field.
Definitions
- Term
Biot–Savart Law
- Meaning
A law that describes the magnetic field generated by an electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.
- Term
Ampere's Circuital Law
- Meaning
A law stating that the line integral of the magnetic field around any closed loop is proportional to the total electric current passing through the loop.
- Term
Permeability of Free Space (μ₀)
- Meaning
A fundamental physical constant representing the ability of a vacuum to permit magnetic field lines to pass through it. Its value is 4π × 10⁻⁷ T·m/A.
- Term
Amperean Loop
- Meaning
An imaginary closed path chosen strategically to apply Ampere's Circuital Law, typically chosen to exploit the symmetry of the magnetic field.
Learning objectives
State and apply Biot–Savart Law to calculate the magnetic field due to a current element.
Derive and apply the magnetic field formulas for a straight current-carrying wire (finite and infinite).
Derive and apply the magnetic field formulas for a circular current loop (at center and on axis).
State and apply Ampere's Circuital Law to calculate magnetic fields for symmetric current distributions.
Calculate the magnetic field inside a solenoid and a toroid.
Determine the direction of magnetic fields using the Right-Hand Thumb Rule.
Formulae
- Name
Biot–Savart Law (Differential Form)
- Note
Calculates the magnetic field dB at a point due to a current element Idl. r̂ is the unit vector from dl to the point, r is the distance. Direction is given by the cross product.
- Expression
dB = (μ₀/4π) * (I dl x r̂ / r²)
- Name
Ampere's Circuital Law
- Note
The line integral of the magnetic field B around any closed loop equals μ₀ × the net current Ienclosed passing through the loop.
- Expression
∮ B ⋅ dl = μ₀ Ienclosed
- Name
Magnetic Field due to an Infinite Straight Current-Carrying Wire
- Note
Where I is the current, 'a' is the perpendicular distance from the wire to the point of observation. Direction is tangential to concentric circles around the wire.
- Expression
B = μ₀I / 2πa
- Name
Magnetic Field at the Center of a Circular Current Loop
- Note
Where I is the current, R is the radius of the loop. For N turns, multiply by N: B = Nμ₀I / 2R. Direction is perpendicular to the plane of the loop.
- Expression
B = μ₀I / 2R
- Name
Magnetic Field on the Axis of a Circular Current Loop
- Note
Where I is the current, R is the radius, and x is the distance from the center along the axis. For N turns, multiply by N: B = (Nμ₀IR² / 2(R² + x²)^(3/2)).
- Expression
B = (μ₀IR² / 2(R² + x²)^(3/2))
- Name
Magnetic Field Inside a Solenoid
- Note
Where n is the number of turns per unit length (n = N/L), and I is the current. The field is uniform and axial inside an ideal long solenoid.
- Expression
B = μ₀nI
- Name
Magnetic Field Inside a Toroid
- Note
Where N is the total number of turns, I is the current, and r is the average radius of the toroid's core. The field is confined within the toroid.
- Expression
B = μ₀NI / 2πr
Prerequisites
Basic understanding of electric current and its direction.
Vector algebra, including cross products.
Calculus (integration) for Biot–Savart Law applications.
Concept of line integrals for Ampere's Circuital Law.
Right-Hand Rule for vector cross products.
Common mistakes
Incorrectly applying the Right-Hand Thumb Rule for direction of magnetic field.
Confusing the distance 'r' in Biot–Savart Law with the radius 'a' or 'R' in specific geometries.
Not considering the number of turns (N) when calculating fields for coils, solenoids, or toroids.
Incorrectly identifying the 'enclosed current' (Ienclosed) for Ampere's Law, especially for multiple wires or loops.
Forgetting the vector nature of Biot–Savart Law and only calculating magnitude.
Keywords
Biot–Savart
Ampere's Law
Magnetic Field
Current Element
Solenoid
Toroid
Right-Hand Rule
Permeability
Amperean Loop
Practice preview
Which of the following expressions correctly represents Biot-Savart Law for the magnetic field dB due to a current element Idl?…
easy
What is the magnitude of the magnetic field B at a distance 'r' from a long straight conductor carrying current 'I'?…
easy
Which of the following statements about the magnetic field inside a long solenoid is INCORRECT?…
medium