Consider a current carrying wire ‘i’ in a specific direction as shown in the above figure. Take a small element of the wire of length ds. The direction of this element is along that of the current so that it forms a vector i ds.
To know the magnetic field produced at a point due to this small element, one can apply Biot-Savart’s Law. Let the position vector of the point in question drawn from the current element be r and the angle between the two be θ. Then,
dB = μ04π i ds sin θr2
Where μ0 is the permeability of free space and is equal to 4π × 10-7 TmA-1.
The direction of the magnetic field is always in a plane perpendicular to the line of element and position vector. It is given by the right hand thumb rule where the thumb points to the direction of conventional current and the other fingers show the magnetic field’s direction.
This can be expressed in terms of vectors as:
d→B = μ04π i →ds ×^rr2
Let us use this law in an example to calculate the Magnetic field due to a wire carrying current in a loop.
Example: Magnetic field of Current Loop
Consider a current loop of radius R with a current ‘i’ flowing in it. If we wish to find the electric field at a distance l from the centre of loop due to a small element ds, we can use the Biot- Savart Law as:
d→B = μ04π i d→s ×^rr2
Consider the current element ids at M which is coming out of plane in the figure. Since r is in plane of the page, the two of them are perpendicular to each other. Furthermore, the magnetic field produced db is also in plane of the page.
dB = μ04π i ds . 1. sin 90⁰r2 = μ04π i dsr2
But from the figure,
R2 + l2 = r2
dB = μ04π i dsR2 + l2
Now, if we consider the diametrically opposite element at N, it produces a field such that its component perpendicular to the axis of the loop is opposite to that of the field produced at M. Thus only the axial components remain. We can divide the loop into diametrically opposite pairs and apply the same logic.
Also note that from figure that
α = θ
∴ cos θ = R√R2 + l2
dB cos θ = μ04π i dsR2 + l2 × R√R2 + l2
The total field will be thus,
B = ∫ μ04π i ds R(R2 + l2)32 = μ04π i R(R2 + l2)32 ∫ ds
B = μ04π i R(R2 + l2)32 × 2πR
B = μ0 i R22(R2 + l2)32
The right hand thumb rule can be used to find the direction of magnetic field.
If there are ‘n’ loops carrying current in the same direction placed very close to one another, can you calculate the field using Biot-Savart’s Law? What if we placed a loop carrying current in the opposite direction right next to the loop in the example we demonstrated? Will it still produce a magnetic field or behave like an electric dipole only this time the dipole is magnetic?
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