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Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET PDF Download

4. MAGNETIC FIELD AT AN AXIAL POINT OF A CIRCULAR COIL

Consider a circular loop of radius R and carrying a steady current i. We have to find out magnetic field at the axial point P, which is at distance x from the centre of the loop.

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Consider an element i Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET of the loop as shown in figure, and the distance of point P from current element is r. The magnetic field at P due to this current element from the equation (1) can be given by,

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

In case of point on the axis of a circular coil, as for every current element there is a symmetrically situated opposite element, the component of the field perpendicular to the axis cancel each other while along the axis add up.

Therefore, Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Here, q is angle between the current element id and , which is everywhere and

sin Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Therefore, B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

or, B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

or, B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET ...(4)   

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

If the coil has N turns, then

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Direction of  Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEETDirection of magnetic field at a point the axis of a circular coil is along the axis and its orientation can be obtained by using the right-hand thumb rule. If the figures are curled along the current, the stretched thumb will point towards the magnetic field.

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Magnetic field will be out of the page for anticlockwise current while into the page for clockwise current as shown in the figure given. Now consider some special cases involving the application of equation (4)

Case I : Field at the centre of the coil

In this case distance of the point P from the centre (x) = 0, the magnetic field

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEETMagnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET  
Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Case II : Field at a point far away from the centre

It means x >> R, B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

4.1 FIELD AT THE CENTRE OF A CURRENT ARC

Consider an arc of radius R carrying current i and subtending an angle f at the centre.

According to Biot-Savart Law, the magnetic field induction at the point P is given by

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Here, dl = Rdθ

Therefore, B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET
   Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

or, B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET ...(5)

It `l' is the length of the circular arc, we have

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET ...(6)

Consider some special cases involving the application of equation (5)

case I : If the loop is semicircular

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

In this case Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET= π, so

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

and will be out of the page for anticlockwise current while into the page for clockwise current as shown in the figure.

Case II. If the loop is a full circle with N turns

In this case Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = 2p, so

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET 
 Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

and will be out of the page for anticlockwise current while into the page for clockwise current as shown in the figure.

Ex.7 Two wire loop PQRSP formed by joining two semicircular wires of radii Rand Rcarries a current i as shown in the figure given below. What is the magnetic field induction at the centre O in cases (A) and (B) ?

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Sol. (a) As the point O is along the length of the straight wires, so the field at O due to them will be zero and hence magnetic field is only due to semicircular portions

Therefore, Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

or, Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET out of the page

(b) Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET into the page

Ex.8 A battery is connected between two points A and B on the circumference of a uniform conducting ring of radius r and resistance R as shown in the figure given below. One of the arcs AB of the ring subtends an angle q at the centre. What is the value of the magnetic field at the centre due to the current in the ring?

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Sol. (a) As the field due to arc at the centre is given by

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Therefore, B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

But (VA - VB) = i1R1 = i2R2

or, i2 = i1Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = i1 Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET [Therefore, R µ L]

i2 = i1Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET [Therefore, L = rq]

Therefore, BRMagnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

i.e., the field at the centre of the coil is zero and is independent of θ.

Ex.9 A charge of one coulomb is placed at one end of a nonconducting rod of length 0.6m. The rod is rotated in a vertical plane about a horizontal axis passing through the other end of the rod with angular frequency 104π rad/s. Find the magnetic field at a point on the axis of rotation at a distance of 0.8 m from the centre of the path.

Now half of the charge is removed from one end and placed on the other end. The rod is rotated in a vertical plane about horizontal axis passing through the mid-point of the rod with the same angular frequency. Calculate the magnetic field at a point on the axis at a distance of 0.4 m from the centre of the rod.

Sol. As the revolving charge q is equivalent to a current

i = qf = q × Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = 1 × Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = 5 × 103 A

Now B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET   Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Therefore, B = 10-7 × Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = 1.13 × 10-3 T

If half of the charge is placed at the other end and the rod is rotated at the same frequency, the equivalent current.

i' = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = qf = i = 5 × 103
  Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

In this case, R' = 0.3 m and x' = 0.4 m

Therefore, B' = 10-7 × Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = 2.3 × 10-3T

5. Solenoid :

(i) Solenoid contains large number of circular loops wrapped around a non-conducting cylinder. (it may be a hollow cylinder or it may be a solid cylinder)

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

(ii) The winding of the wire is uniform direction of the magnetic field is same at all points of the axis.

(iii) Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET on axis (turns should be very close to each others).

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET ...(7)

where n : number of turns per unit length.

cosθ1Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET ; cosβ = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET = -cosθ2   Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

Derivation :

Take an element of width dx at a distance x from point P. [point P is the point on axis at which we are going to calculate magnetic field. Total number of turns in the element dn = ndx where n : number of turns per unit length]

dB = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET

B = Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET            

The document Magnetic Field on the Axis of a Circular Current Loop | Physics Class 12 - NEET is a part of the NEET Course Physics Class 12.
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FAQs on Magnetic Field on the Axis of a Circular Current Loop - Physics Class 12 - NEET

1. What is the formula to calculate the magnetic field on the axis of a circular current loop?
Ans. The formula to calculate the magnetic field on the axis of a circular current loop is given by the equation B = (μ0 * I * R^2) / (2 * (R^2 + x^2)^(3/2)), where B is the magnetic field, μ0 is the permeability of free space, I is the current flowing through the loop, R is the radius of the loop, and x is the distance from the center of the loop to the point on the axis where the magnetic field is being calculated.
2. How does the magnetic field on the axis of a circular current loop vary with distance?
Ans. The magnetic field on the axis of a circular current loop decreases with increasing distance from the center of the loop. As the distance (x) increases, the magnetic field strength decreases inversely proportional to the cube of the sum of the radius of the loop (R) and the distance squared (x^2). This means that the magnetic field becomes weaker as you move further away from the loop.
3. Does the direction of the magnetic field on the axis of a circular current loop change with distance?
Ans. No, the direction of the magnetic field on the axis of a circular current loop remains the same regardless of the distance from the loop. The magnetic field is always perpendicular to the plane of the loop and follows the right-hand rule. If the current flows clockwise when viewed from above, the magnetic field points in the counterclockwise direction on the axis of the loop, and vice versa.
4. How does the current flowing through the loop affect the magnetic field on its axis?
Ans. The magnetic field on the axis of a circular current loop is directly proportional to the current flowing through the loop. Increasing the current will result in a stronger magnetic field, while decreasing the current will result in a weaker magnetic field. The relationship between the magnetic field and the current is linear, meaning that doubling the current will double the magnetic field strength.
5. Can the magnetic field on the axis of a circular current loop be zero?
Ans. No, the magnetic field on the axis of a circular current loop cannot be zero. The magnetic field is always present and its strength depends on the variables such as the current, radius of the loop, and distance from the center of the loop. However, it is possible for the magnetic field to be very close to zero if the current is extremely small or the distance from the loop is very large.
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