Question

In a circular loop conductor, what is the value of magnetic field at the centre?

Answer 1

Answer:

The magnetic field strength at the center of a circular loop is given by B=μ0I2R(at center of loop), where R is the radius of the loop. RHR-2 gives the direction of the field about the loop.

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Answer 2

Explanation:

We can use the Biot-Savart law to find the magnetic field due to a current. We first consider arbitrary segments on opposite sides of the loop to qualitatively show by the vector results that the net magnetic field direction is along the central axis from the loop. From there, we can use the Biot-Savart law to derive the expression for magnetic field.

Let P be a distance y from the center of the loop. From the right-hand rule, the magnetic field dB⃗ at P, produced by the current element Idl⃗ , is directed at an angle θ above the y-axis as shown. Since dl⃗ is parallel along the x-axis and rˆ is in the yz-plane, the two vectors are perpendicular, so we have

dB=μ04πIdlsinθr2=μ04πIdly2+R2

where we have used r2=y2+R2.

Now consider the magnetic field dB⃗ ′ due to the current element Idl⃗ ′, which is directly opposite Idl⃗ on the loop. The magnitude of dB⃗ ′ is also given by Equation, but it is directed at an angle θ below the y-axis. The components of dB⃗ and dB⃗ ′ perpendicular to the y-axis therefore cancel, and in calculating the net magnetic field, only the components along the y-axis need to be considered. The components perpendicular to the axis of the loop sum to zero in pairs. Hence at point P:

B⃗ =jˆ∫loopdBcosθ=jˆμ0I4π∫loopcosθdly2+R2.

For all elements dl⃗ on the wire, y, R, and cosθ are constant and are related by

cosθ=Ry2+R2−−−−−−√.

Now from Equation, the magnetic field at P is

B⃗ =jˆμ0IR4π(y2+R2)3/2∫loopdl=μ0IR22(y2+R2)3/2jˆ

where we have used ∫loopdl=2πR. As discussed in the previous chapter, the closed current loop is a magnetic dipole of moment μ⃗ =IAnˆ. For this example, A=πR2 and nˆ=jˆ, so the magnetic field at P can also be written as

B⃗ =μ0μjˆ2π(y2+R2)3/2.

By setting y=0 in Equation, we obtain the magnetic field at the center of the loop:

B⃗ =μ0I2Rjˆ.

This equation becomes B=μ0nI/(2R) for a flat coil of n loops per length. It can also be expressed as

B⃗ =μ0μ⃗ 2πR3.

If we consider y≫R in Equation, the expression reduces to an expression known as the magnetic field from a dipole:

B⃗ =μ0μ⃗ 2πy3.

The calculation of the magnetic field due to the circular current loop at points off-axis requires rather complex mathematics, so we’ll just look at the results. The magnetic field lines are shaped as shown in Figure. Notice that one field line follows the axis of the loop. This is the field line we just found. Also, very close to the wire, the field lines are almost circular, like the lines of a long straight wire.

Figure shows the magnetic field lines of a circular current loop. One field line follows the axis of the loop. Very close to the wire, the field lines are almost circular, like the lines of a long straight wire.

Sketch of the magnetic field lines of a circular current loop.