# Helmholtz Coils

Helmholtz coils are used to produce a region of constant applied magnetic field.   In some experiments they are used to counter the Earth's magnetic field.  This pre-lab exercise will investigate the geometric properties of Helmholtz coils.

The magnetic field on the axis of a circular current loop is found by an application of the Biot-Savart law to be

where R is the radius of the loop, I is the current in the loop and the center of the loop at x = +h.  The loop is in a plane perpendicular to the x-axis.   See your text, for example, for the derivation of this relation.   In order to simplify the equations to be entered, assume that moI/2 =1.  Below is a plot of Bx for a single loop of radius 0.8 units and located at x = +1 unit.

f(x,t)=

Place the mouse cursor inside the plot and hold down the left button to show the coordinates.  What is the position of the peak?  Use this value in the formula to get the y-coordinate of the peak. Does it yield the correct value for y?

Click here to Start the Exercise.    The principle of superposition can be used to find the magnitude of the magnetic field for two identical loops which are coaxial and separated by a distance 2h.  Study the equation in the box above.

The radius of the loops is represented by the variable t.   The current value for t is shown in the upper left hand corner of the graph. The step>> button increments t in 0.1 unit increments.  Vary the radius of the loops.  From the equation it is seen that the initial value for the coil separation is 2 units. For some values of t, the function will go off the screen.

1. For what value of t is the magnitude of the x-component of the field between the loops the strongest AND most uniform?  Over what distance is it fairly constant?
2. Change the positions of the loops to be x = +/- 0.75 (a separation of 1.5 units) and repeat the above the procedure.  What ratio (Helmholtz ratio) between the coil radius and separation gives the strongest AND most uniform field?
3. Change the separation to a different value. Does your prediction for the Helmholtz ratio in #2 still hold?
4. Place the coils at x = +/- 0.5.  Make the current direction in the coils to be opposite.  Describe the field between the loops for ratios around the Helmholtz ratio.