AIM - To study the variation of magnetic field with distance along the axis of Helmholtz galvanometer and to estimate the radius of the coil.

APPARATUS - Helmholtz galvanometer, magnetometer compass box, ammeter (measure mA), battery eliminator, rheostat, commutator, plug key, connecting wires.

THEORY- The term Helmholtz coils refers to a device for producing a region of nearly uniform magnetic field. It is named in honor of German physicist Hermann von Helmholtz. It consists of two identical circular magnetic coils that are placed symmetrically parallel to each other and on a common axis, z- axis. The rings have radius r and they are separated by a distance equal to or slightly larger than r. Each coil carries an equal electrical current flowing in same direction.
[pic]
Figure 1.Helmholtz coils

The first step to calculate the field of a pair of Helmholtz coil is to calculate magnetic field intensity F produced by each ring.
If a current (I) is allowed to flow through a wire of length ([pic]), and the wire is bent into an arc of radius r, then the magnetic field intensity (F) at center of the arc is
[pic] (1) where [pic]= permeability of free space (8.854 x 10-12 F/m)
For a circular coil of n turns we substitute l=2(rn in equation (1)
[pic] (2)
Now, substituting the value of [pic]in equation (2)
The magnetic field produced by each ring is given by
[pic] (3)

The magnetic field at any point on axis at a distance (x) from center of coil is

[pic] (4)

The rate of variation of magnetic field.
[pic]

Figure2. Magnetic field generated by a coil with radius (r) =1m.
[pic]

Figure3. Magnetic field generated by a pair of Helmholtz coils

Therefore,
[pic]
From which, [pic] ,if [pic]or [pic]constant.
Thus at point [pic] from center of coil, [pic]constant

We observe that in figure3, the rate of increase of field due to one coil at midpoint between the coils is equal to the rate of decrease of field due to the other at the same point. Therefore if one moves away along the axis from the midpoint, any diminution in the intensity of the field due to one coil is compensated by the increase in the field due to the other so that the field between the coils is practically uniform.

The coil is placed in the magnetic meridian, the magnetic field due to the current I flowing in the coil is perpendicular to H (Horizontal component of earth’s magnetic field). Thus the magnetic needle is acted upon by two uniform magnetic fields F and H at right angles to each other.
The magnetic needle will make an angle[pic] with H in the equilibrium position.

According to tangent law:
F = H tan[pic]
Therefore
[pic] (5)
Note: tan[pic]α I tan [pic]α 1 r3

Magnetometer: The magnetic compass box used in this experiment is called Magnetometer. The red point in the magnetometer corresponds to North direction (Red is analogous to positive terminal of electrical circuit.). Thus in absence of magnetic field the needles are in east west direction and while performing the experiment in order to avoid to the Earth’s horizontal magnetic field the bench of the Helmholtz galvanometer should be kept in east-west direction.

PROCEDURE- 1. The Helmholtz coils should be parallel to themselves and perpendicular to the bench and at a distance equal to [pic] on either side from center of the bench. 2. Magnet compass box is kept at kept at the center of the sliding bench, such that magnetic needle is at the center of the coils. 3. The bench of the Helmholtz galvanometer should be kept in east-west direction 4. Base of the coil is leveled with the help of spirit level and leveling screws. 5. Connections are made as shown in fig. 4 using say 50 turns of the coil and taking care that out of the four terminals provided on the commutator K any two diagonally opposite terminals are joined to the galvanometer and the other two to the battery through rheostat. [pic] Figure4. Circuit diagram

6. Adjust the current in the coil with the help of rheostat such that the deflection in the magnetic needle is of the order of 45( at center of bench for both direct and reverse current.
[1]

7. Now move the compass box through 2 cm and note the deflection in east and west direction of magnetometer for direct and reverse current respectively. 8. Continue to take readings till +/-15( deflection is obtained in the compass box with respect to 45(. 9. Repeat the procedure for other side. 10. Calculate mean [pic] and find tan[pic] according to the observation table. 11. Plot a graph taking x on x-axis and tan[pic] on y-axis respectively for each side. Mark the points of inflection on the curve. The distance between the two points will be the diameter of the coil. 12. The circumference of the coil can be measured by a thread and its radius can be calculated to verify the value obtained from the graph

OBSERVATIONS-
| |
| | |
|S.n. |Position of the |
| |needle on one of the |
| |scale. (Distance of |
| |Compass box from |
| |center of coil) |
| |x (cm) |
| | |DeDeflection in the needle when it is on one side of bench |
| | |Current one way |Current reversed | Mean |Tan [pic] |
| | | | |[pic] | |
| | | | |(degree) | |
| | |[pic] |[pic] |[pic] |[pic] | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | |East |West |East |West | | |
| | |end of |end of |end of |end of | | |
| | |needle |needle |needle |needle | | |
|1. |0 | | | | | | |
|2. |2 | | | | | | |
|3. |4 | | | | | | |
|4. |6 | | | | | | |
|5. |8 | | | | | | |
|6. |10 | | | | | | |
|7. |12 | | | | | | |
|8. |14 | | | | | | |
|9. |16 | | | | | | |
|10. |18 | | | | | | |
|11. |20 | | | | | | |
|12. |22 | | | | | | |
|13. |24 | | | | | | |
|14. |26 | | | | | | |
|15. |28 | | | | | | |
|16. |30 | | | | | | |

| |
| | |
|S.No. |Position of the |
| |needle on one of the|
| |scale. (Distance of |
| |Compass box from |
| |center of coil) |
| |x (cm.) |
| | |DeDeflection in the needle when it is on other side of bench |
| | |Current one way |Current reversed | Mean |Tan[pic] |
| | | | |[pic] | |
| | | | |degree | |
| | |[pic] |[pic] |[pic] |[pic] | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | |East |West |East |West | | |
| | |end of |end of |end of |end of | | |
| | |needle |needle |needle |needle | | |
|1. |0 | | | | | | |
|2. |2 | | | | | | |
|3. |4 | | | | | | |
|4. |6 | | | | | | |
|5. |8 | | | | | | |
|6. |10 | | | | | | |
|7. |12 | | | | | | |
|8. |14 | | | | | | |
|9. |16 | | | | | | |
|10. |18 | | | | | | |
|11. |20 | | | | | | |
|12. |22 | | | | | | |
|13. |24 | | | | | | |
|14. |26 | | | | | | |
|15. |28 | | | | | | |
|16. |30 | | | | | | |

GRAPH:

[pic]west[pic] east[pic]

CALCULATION-

Circumference of the coil as obtained by a thread and meter scale = …..cm.

Radius of the coil, as obtained from the graph = distance between the points A and B.

Radius of the coil, as obtained from measurement = [pic] (6)

MAXIMUM PROBABLE ERROR:- This is obtained by taking logarithmic differentiation of equation (6)

RESULT: - 1. The variation in the magnetic field with distance, along the axis of the given coil is as shown in the graph. 2. Radius of the coil = __________cm. as obtained from the graph and _______________cm. as obtained from measurement.

PRECAUTIONS- 1. Connections should be clean and tight. 2. Circuit should be properly connected and checked before turning it “ON”. No. of turns should be equal in both coils. 3. Plug key should be used in circuit and it should not be closed while making connections or taking reading. 4. The coil should be adjusted properly in the magnetic meridian. 5. The apparatus should be at considerable distance from current carrying conductors and magnetic materials. 6. The positive marked terminal of the ammeter should be always connected to positive terminal of battery. 7. While taking readings there can be error due to parallax which should be avoided. 8. Readings at both ends of the pointer should be taken.

SOURCES OF ERROR- 1. Connections might not be tight. 2. Magnetic needle might not be pivoted at center. 3. Galvanometer coil might not be exactly in magnetic meridian.

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[1] 45( is an ideal condition and students should try to approach as close as possible

Students are not required to level the base of coil.