 # Magnetic Field Due to a Current Element, Biot-Savart Law We all know that magnetic field is produced by the motion of electric charges or electric current. Biot-Savart law gives this relation between current and magnetic field. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.

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### Suggested Videos        ## Biot-Savart Law

A small current carrying conductor of length dl, carrying a current I is an elementary source of magnetic field. The force on another similar conductor can be expressed conveniently in terms of magnetic field dB due to the first. The dependence of magnetic field dB on the current I, on size and orientation of the length element dl and on distance r was first guessed by Biot and Savart.

## Mathematical Representation

The Biot-Savart’s law gives the magnetic field produced due to a current carrying segment. This segment is taken as a vector quantity known as the current element.

Consider a wire carrying a current I in a specific direction as shown in the figure. Take a small element of the wire of length dl. The direction of this element is along that of the current so that it forms a vector Idl. If we want to know the magnetic field produced at a point due to this small element, then we can use the Biot-Savart’s Law.

The magnitude of the magnetic field dB at a distance r from a current carrying element dl is found to be proportional to I and to the length dl. And is inversely proportional to the square of the distance |r|. The direction of the Magnetic Field is perpendicular to the line element dl as well as radius r. (Source: learnCBSE)

Thus the vector notation is given as, dB α Idl × r /  r3 = (μ/ 4π ) × (Idl × r / r3),where μ0/4π is a constant of proportionality. The above expression holds when the medium is a vacuum. Therefore the magnitude of this field is:

|dB| =  (μ0 / 4π) × (Idl sinθ / r2)

## Similarities And Differences Between Biot-Savart Law And Coulomb’s Law

### Similarities

• Both magnetic and electric fields at a point are inversely proportional to the square of the distance between the field source and the point in question.
• Electric field due to a point charge (Coulomb’s law) is: E = (1/4πƐo) × (q/r2)
• Magnetic field due to a moving charge (Biot-Savart law) is: B = (μo/4π) × Idl (sinθ)/r2

### Differences

• The source of the electrostatic field is scalar in nature. Whereas, the source of the magnetic field, which is the current element (Idl), is a vector in nature.
• The electric field always acts along the plane containing distance (r) between a point charge and the point where the electric field is to be calculated. But, the magnetic field acts in the plane perpendicular to the plane of distance(r) between the current element and the concerned point.
• Magnetic field depends on both the angle (θ) between the current element (Idl) and the line joining the point and current element. However, the electric field doesn’t depend on the angle (θ). (Source: ExamFear)

The first diagram shows the electric field (E) due to a point charge (q)The second diagram shows the magnetic field (B) due to the current carrying wire.

## Solved Examples For You

Question: A circular coil is of 10 turns and radius 1m. If a current of 5A flows through it, calculate the field in the coil from a distance of 2m.

A) 314.16 × 10-7 T     B) 341.61 × 10-7 T       C) 200 × 10-7 T      D) 314.16 × 10-10 T

Solution: Given: No. of turns n = 10, Current I = 5A, length l = 2m, radius r = 1m

The Biot-Savart law formula is given by,

B = (μo / 4π) × (2πnI / r)

Therefore, B = (μo / 4π) × (2 × π × 10 × 5 / 1)

B  = 314.16 × 10-7 T

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