AP PHYSICS

DR.RIGOL

__DETERMINATION OF THE
MAGNETIC FIELD INSIDE A HELMHOLTZ COIL__

__INTRODUCTION __:

The Helmholtz coils is a combination of two circular coils of
equal radius that are mounted in parallel along a common axis. The condition
for these coils is that the radius of the coils is equal to the distance
between the coils.

We made the Helmholtz coils using
enameled wire with diameter 0.9 mm. The 60 loops in each coil were rolled with
a radius 0.059 m.

To determine the magnetic
field inside the Helmholtz coil, we determine the force acting on a wire
conducting a current located inside the magnetic field created by the Helmholtz
coils. The conducting current wire is hanging from one arm of a balance. The
balance is calibrated to null when no current is going through the wire. When a
current is present, the Lorentz force acting on the wire will alter the state
of equilibrium in the balance. To restore the point of equilibrium some
additional weight is added to the right arm of the balance. The added amount of
weight will be equal to the magnitude of the magnetic force acting on the wire.
Knowing the magnetic force and the current going through the wire, the Lorentz
force formula can be used to calculate the magnitude of the magnetic field.
This magnitude of the magnetic field determined in the experiment, is then
compared to the calculated value using the Biot-Savart
Law.

**MAGNETIC FORCE
ACTING ON A WIRE WITH ELECTRIC CURRENT**:

When a wire with length ** L** inside a magnetic field with
magnetic induction

In this expression L is a vector that has the direction of
the current and the magnitude of the wire length. As a consequence of the
vector product, the direction of the force will be perpendicular to the
direction of the current, and to the direction of the magnetic field.

**HELMHOLTZ COILS**:

Now, I will
derive the expression of the magnetic field created on the axes of symmetry at
the middle point between the two coils. I will start with the Biot-Savart law that tells us that the magnetic field
created by an element of current I at the distance r is given by the expression:

d

In this expression, _{0}
is the permeability of the vacuum [_{0 }= (T.m/A)], and ** **is the element of current creating the magnetic field at the distance**. **

The element of magnetic
field created by Idl is shown in the fig. above. This
vector can be represented by one x-component that will be looking to the left,
and a y-component that will be looking down. When we integrate around the coil,
the final sum of all y-components will be zero and the total magnetic field
will be the x-component that will be looking to the left. That is why a factor cos(j) will appear in the above formula.

dB

the relationship between r and R can be
found from the graph:

r =

The integral of dl will
be 2πR, so the value for the magnitude magnetic field B will be:

B = [4/(125)^{1/2}]^{
}µ_{0} I/R^{ }

If we consider that the
coil has N loops, and that we have two coils, then the final result will be:

B = (4/5)^{3/2 µ}_{0}
NI/R^{ }

**EXPERIMENTAL VALUES FOR
B**:

Our Helmholtz coils
have 60 loops and the radius is R=0.059 m, so the *expected* value for B will be given by the expression:

B_{e}= 9.1×10^{-4 }I (T)

To find the observed
value of B (B_{o}), we use the expression of the Lorentz force:

B_{o} =

To increase the effect
of the magnetic force, instead of using a simple wire we use a coil with 40
loops. This coil is shaped as a square with mean side size 0.070±0.005 m.

This coil is hanging
from the balance and inside the Helmholtz coil in such a way that the top is
outside and the bottom is in the center of the Helmholtz coils. Observe that
the current going through this coil (I_{c})is, in general, different from the current going through
the Helmholtz coils (I_{HC}).

# |
I |
I |
F(N) |
B |
B |
[B |

1 |
2 |
3 |
0.014 |
1.7 |
1.8 |
-0.1 |

2 |
2 |
4 |
0.021 |
1.9 |
1.8 |
0.1 |

3 |
2 |
5 |
0.026 |
1.9 |
1.8 |
0.1 |

4 |
3 |
3 |
0.024 |
2.9 |
2.7 |
0.2 |

5 |
3 |
4 |
0.031 |
2.8 |
2.7 |
0.1 |

6 |
3 |
5 |
0.036 |
2.6 |
2.7 |
-0.1 |

7 |
4 |
3 |
0.028 |
3.3 |
3.6 |
-0.3 |

8 |
4 |
4 |
0.038 |
3.4 |
3.6 |
-0.2 |

9 |
4 |
5 |
0.048 |
3.4 |
3.6 |
-0.2 |

10 |
5 |
5 |
0.062 |
4.4 |
4.6 |
-0.2 |