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End-on (or) Tan A position- Deflection magnetometer

End-on (or) Tan A position- Deflection magnetometer
The magnetic field at a point along the axial line of a bar magnet is perpendicular to the horizontal component of Earth?s magnetic field. If a magnetometer and a bar magnet are placed in such way that this condition is satisfied, then this arrangement is called Tan A position.

 

Deflection magnetometer

 

Deflection magnetometer consists of a small magnetic needle pivoted on a sharp support such that it is free to rotate in a horizontal plane. A light, thin, long aluminium pointer is fixed perpendicular to the magnetic needle. The pointer also rotates along with the needle (Fig.).

There is a circular scale divided into four quadrants and each quadrant is graduated from 0o to 90o. A plane mirror fixed below the scale ensures, reading without parallax error, as the image of the pointer is made to coincide exactly with pointer itself. The needle, aluminium pointer and the scale are enclosed in a box with a glass top. There are two arms graduated in centimetre and their zeroes coincide at the centre of the magnetic needle.


 

End-on (or) Tan A position- Deflection magnetometer

 

The magnetic field at a point along the axial line of a bar magnet is perpendicular to the horizontal component of Earth?s magnetic field. If a magnetometer and a bar magnet are placed in such way that this condition is satisfied, then this arrangement is called Tan A position.

 

To achieve this, the arms of the deflection magnetometer are placed along East-West direction (i.e)  perpendicular to the magnetic meridian. The bar magnet is placed along East - West direction (i.e) parallel to the arms, as shown in the Fig.


When a bar magnet of magnetic moment M and length 2l is placed at a distance d from the centre of the magnetic needle, the needle gets deflected through an angle ? due to the action of two magnetic fields.

 

(i) the field B due to the bar magnet acting along its axis and

 

(ii) the horizontal component of Earth?s magnetic field Bh.

 

The magnetic field at a distance d acting along the axial line of the bar magnet,

B = ?0/ 4 π . 2 Md/(d2-l2) 2

According to Tangent law,

B = Bh tan θ

?0/ 4 π . 2 Md/(d2-l2) 2 = Bh tan θ

 

Comparison of magnetic moments of two bar magnets

 

(i) Deflection method

 

The deflection magnetometer is placed in Tan A position (Fig.). A bar magnet of magnetic moment M1 and length 2l1 is placed at a distance d1 from the centre of the magnetic needle, on one side of the compass box. Since, the sensitivity of the magnetometer is more at 45o, the distance of the bar magnet should be chosen such that the deflection lies between 30o and 60o. The readings corresponding to the ends of the aluminium pointer are noted as θ 1 and θ 2. The magnet is reversed pole to pole and kept at the same distance. Two more readings θ3 and θ4 are noted. By placing the magnet on the other side of the compass box at the same distance, four more readings θ 5, θ 6, θ 7 and θ 8 are noted as above. The mean of the eight readings gives a value θI.


The experiment is repeated as above for the second bar magnet of magnetic moment M2 and length 2l 2 by placing at a

distance d2. Now the mean of the eight readings gives a value of θII.

 

Applying tangent law, for the first magnet,

?0/4 π  . 2M1d1/(d12-l12)2 = Bhtan θI  ?????.(1)

 

and for the second magnet.

?0/4 π  . 2M2d2/(d22-l22)2 = Bhtan θII  ?????.(2)

 

From the above equations (1) and (2), we get

M1/M2 =  [ (d12 ? l12)2 tan θI  ] / [ (d22 ? l22)2 tan θII  ]

 

Special case

If the magnets are placed at the same distance, then d1 = d2 = d

M1/M2 =  [ (d12 ? l12)2 tan θI  ] / [ (d22 ? l22)2 tan θII  ]

 

In addition, if l1 and l2 are small compared to the distance d

Then M1/M2 = tan θI / tan θI

 

(ii) Null deflection method

The deflection magnetometer is placed in Tan A position (Fig.). A bar magnet of magnetic moment M1 and length 2l1 is placed on one side of the compass box at a distance d1 from the centre of the magnetic needle. The second bar magnet of magnetic moment M2 and length 2l2 is placed on the other side of the compass box such that like poles of the magnets face each other. The second magnet is adjusted so that the deflection due to the first magnet is nullified and the aluminium pointer reads 0o - 0o. The distance of the second magnet is x1. The first magnet is reversed pole to pole and placed at the same distance d1. The second magnet is also reversed and adjusted such that the aluminium pointer reads 0o - 0o. The distance of the second magnet is x2.

 

The experiment is repeated by interchanging the magnets. Two more distances x3 and x4 are noted. The mean of x1, x2, x3 and x4 is taken as d2.

 

As the magnetic fields due to the two bar magnets at the centre of the magnetic needle are equal in magnitude but opposite in direction,

 

(i.e) B1 = B2

M1/M2 =  [ (d12 ? l12)2 d2  ] / [ (d22 ? l22)2 d1  ]

If the bar magnets are short, l1 and l2 are negligible compared to the distance d1 and d2

M1/M2 = d13 / d23


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