**Biot-Savart law**

Biot-Savart law is the basic equation that gives us the magnetic field due to electric current in a conductor.

The magnetic field due to a current element dl at a point P with a distance r from dl is

Vector of dB = Vr(dB) = [1/4πε

_{0}c²]i[Vr(dl)×Vr(r)]/r³ ...(35.1)

where c = speed of light

i = current

Vr(dl) = length vector of the current element

Vr(r) = ector joining the current element to the point P where we are finding magnetic field

[1/ε

_{0}c²] is written as μ

_{0}and is called the permeability of vacuum.

The value of μ

_{0}is 4π*10

^{-7}

Vector of dB = Vr(dB) = [μ

_{0}i/4π][Vr(dl)×Vr(r)]/r³ ... (35.2)

The magnitude of the magnetic field is

dB = [μ

_{0}idl sinθ /4πr²]

Where θ is the angle between current element and the vector joining current element and the point P.

The direction of the field is perpendicular to the plane containing the current element and the point.

**Magnetic field due to current in a straight wire at a point P with a distance d from it.**

B = [μ

_{0}i/4πd][cosθ

_{1}- cosθ

_{2}]

θ

_{1}and θ

_{2}are the values of θ corresponding to the lower end and the upper end respectively of the straigth wire.

See the Fig 35.3 to remember the method of measuring θ.

If point P is on a perpendicular bisector of the wire.

θ

_{1}and θ

_{2}are equal. If the length of the wire is α and distance of the point P is d from the wire

cosθ

_{1}= α/SQRT(α²+ 4d²)

cosθ

_{2}= - α/SQRT(α²+ 4d²)

θ

_{1}is anlge withlower end of the wire.

θ

_{2}is anle with the upper end of the wire.

B = [μ

_{0}iα]/[2πdSQRT(α²+ 4d²)]

If the straight wire is a very long one

θ

_{1}= 0 and θ

_{2}is equal to π. So

B = μ

_{0}i/2πd ... (35.6)

**Magnetic field lines**

Magnetic field lines are similar to electric field lines. A tangent to a magnetic field line gives the direction of the magnetic field existing at that point. For a long straight wire, the field lines are circles with their centres on the wire.

**Force between parallel wires carryng current**

If the two wires are treated as long straight wires carrying current i

_{1}(W1) and i

_{2}(W2)

Field on an element on the wire carrying current i

_{2}is

B = μ

_{0}i

_{1}/2πd

Magnetic force on the element

dF = i

_{2}dlμ

_{0}i

_{1}/2πd

So the force per unit length of the wire

W2 due to the wire W1 is

dF/dl = i

_{2}μ

_{0}i

_{1}/2πd

= μ

_{0}i

_{1}i

_{2}/2πd ...(35.7)

Same amount of force is applied by W2 on unit length of W1.

If both the wires carry current in same direction they attract each other. If they carry current in opposite directions, they repel each other.

**Definition of ampere**

If two parallel, long wires, kept 1 m apart in vacuum, carry equal currents in the same directin and there is a force of attraction of 2*10

^{-7}newton per metre of each wire, the current in each wire is said to be 1 ampere.

**Field due to a circular current**

**Field at the centre**

Radius of circular loop = a

current in the loop = i

B = μ

_{0}i/2a

**Field at an axial point due to a circular conductor**

B = μ

_{0}ia²/2(a²+d²)^(3/2)

where

a = radius of the circular conductor

d is the distance of the point from the centre of the circular conductor

If the field is far away from the centre

d>>a

B = 2μ

_{0}πia²/4πd³

As πia² is magnetic dipole moment of circular conductor (μ)

B = 2μ

_{0}μ/4πd³ ... (35.9)

**Ampere's law**

The circulation of Line integral of Vr(B).Vr(dl)of the resultant magnetic field along a closed, plane curve is equal to μ

_{0}times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant.

Line integral of Vr(B).Vr(dl) = μ

_{0}i ... (35.10)

You can derive the magnetic field at a point due to current in a long straight wire

using Ampere's law and verify that the formula is the same as the one derived by using Biot Savart Law.

Ampere's law can be derived from Biot-Savart law and Biot-Savart law from Ampere's law.

Ampere's law is useful under certain symmetrical conditions.

**Finding magnetic field at a point due to a long, straight current using Ampere's law**

**Solenoid**

A solenoid is an insulated wire wound closely in the form of a helix. The length of the solenoid is large compared to its radius of its loop.

The magnetic field inside a very tightly wound long solenoid is uniform everywhere and it zero outside it.

B = μ

_{0}ni ...(35.11)

n= number of turns per unit length

**Toroid**

If a nonconducting ring is taken and a conducting wire is wound closely around it we get a toroid.

In a toroid magnetic field is

B = μ

_{0}Ni/2πr

Where N = number of turns in the toroid

r = is the distance of point P (where we have to find the magnetic field) from the centre of the toroid.

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