Explain Biot-Savart law for magnetic field produced due to an element of a current carrying conductor and use it to define the SI unit of current ampere.

Biot and Savart presented a law for magnetic field produced due to an electric current element in the following form: The intensity of magnetic due to an electric current element I dl → at a point with position vector r ⃗ with respect to electric current element is given by the formula as follows: d B → = μ 0 4 π Idl ⃗ ⊗ r ⇀ r 2 Here, I dl → is the current element, that is the product of current and the line element of the conductor of very small length, μ 0 is the magnetic permeability of vacuum whose value is 4 π × 10 − 7 Tm / A and r → is the unit vector, that is r → = r ¯ r 2 Let us consider a long wire carrying a current I and also consider a point p in the space. Let us also consider an infinitely small length of the wire dl at a distance r from the point P as shown. r ¯ is a distance-vector which makes an angle θ with the direction of current in the infinitesimal portion of the wire. The magnetic field density at point P due to that infinitesimal length dl of the wire is directly proportional to current carried by this portion of the wire. As the current through that infinitesimal length of wire is the same as the current carried by the whole wire itself, we can write, because the current elements are continuous, the vector addition can be written as line integral in the following way B → = ∫ d B → = μ 0 4 π ∫ 1 × d l → × r ¯ r 2 Above is the integral form of the Biot-Savart’s law. The unit of current is ampere. Through Biot-Savart’s law, it can be defined as when 1 Tesla magnetic field is produced from a current carrying conductor at a distance of 1m then 1 ampere current is produced.

Find the expression for intensity of magnetic field produced at a point on the axis of a curret carrying circular coil.

In the above diagram, Let O be the centre of the ring, R be the radius of the ring. The point P lie outside the ring on the axis, let the distance between the P and o be x, where we have to calculate the magnetic field. Let dB → be the magnetic field due to a small length dl of the ring, As per bio-sievert rule, dB → = μ 0 IdI 4 π r 2 We know that r = x 2 + R 2 So db → = μ 0 IdI 4 π x 2 + R 2 This field have vertical dB y → = dB → sinθ and horizontal component dB → x = dB → cos θ Vertical component will cancel out each other, only horizontal components are responsible, so , B → = dB → 1 + dB 2 → + . . . . . dB → = μ 0 IdIsinθ 4 π R 2 + x 2 dB → = μ 0 Ix 2 4 π R 2 + x 2 3 2 The magnetic field due to the circular current loop of radius a at a point which is a distance R away, and is on its axis, So B → = μ 0 Ix 2 2 R 2 + x 2 3 2

Establish the relation among electromagnetic force, terminal voltage and internal resistance of a cell.

Let V be the terminal voltage and E be the emf. Having a load R and current flowing I and internal resistance r can be calculated as – I = E R + r And the potential difference across the terminal voltage calculated as: V = I . R = RE R + r