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## Objectives

- To determine  the relationship between magnetic field and a coil's current and number of turns
- To determine  how the magnetic field in a solenoid relates to theory
- To determine  the earth's magnetic field

## Introduction

When an electric current flows through a wire, a magnetic field is produced around the wire. The magnitude and direction of the field depends on the shape of the wire and the direction and magnitude of the current through the wire. If the wire is wrapped into a loop, the field near the center of the loop is perpendicular to the plane of the loop. When the wire is looped a number of times to form a coil, the magnetic field at the center increases. A solenoid is formed when many turns of wire are wrapped around a tube. Solenoids and coils are used in electronic circuits or as electromagnets.

In this lab, you will examine how the magnetic field is related to both the number of turns in a coil and the current through the coil with a magnetic field sensor. A complication that must be considered is that the sensor will also detect the earth's field and any local fields due to electric currents or some metals in the vicinity of the sensor.

A Slinky will be used to study a solenoid. By inserting a magnetic field sensor between the coils of the Slinky you can study how the field varies in different parts of the solenoid. You will also measure the permeability constant a fundamental constant of physics.

 Figure 1

## Theory - Magnetism

The magnetic field due to a current - carrying element of wire is given by the Biot-Savart Law. The magnitude of the magnetic field at a point P due to a current carrying element of wire of length dl is:

 $dB = \frac{\mu _o I dl sin(\theta)}{4\pi r^2}$ (1)

Constant $$\mu_o$$ is the permeability of vacuum.

The direction of the magnetic field, $$\mathbf{d\vec{B}}$$, is given by a right-hand rule, perpendicular to both  $$\mathbf{d\vec{l}}$$ and $$\mathbf{\vec{r}}$$. Grasp the wire in your right hand with your thumb in the direction of the current. The direction of the magnetic field is in the direction in which the curled fingers are pointing. For the example shown, the magnetic field is out of the page.

The total magnetic field $$\mathbf{\vec{B}}$$ due to a circuit of any shape may be found by adding up (integrating) all the contributions by Equation 1 over the circuit.

### Circular Loop

For a single circular loop of radius R carrying a current I, the magnetic field at the center is especially easy to determine because distance r and angle $$\theta$$ are constant for all $$\mathbf{d\vec{l}}~(r = R,\theta = 90^o)$$. Without requiring integration, the result

 $B=\frac{\mu_oI}{2R}$ (2)

For the situation shown, the magnetic field at the center C is out of the page.

Figure 2

### Square Loop

For a single square loop of side length L, the magnetic field at the center can be found in a similar way using integration.. The result is:

 $B=\frac{2\sqrt{2}\mu_o I}{\pi L}$ (3)

For the situation shown, the magnetic field at C is out of the page. The magnetic field at the center of a square loop is about 90% of a circular loop with the diameter equal to the length of square loop side.

Figure 3

### Solenoid

The field inside a solenoid of a circular cross section where n is the number of turns per length is:

 $B=\mu_o nI$ (4)

## Theory - Magnetic Field Sensor

The sensor uses a Hall effect transducer that produces a voltage that is linear with magnetic field. The sensor measures the component of the magnetic field that is perpendicular to the flat sensor covered with black heat-shrink tubing. Maximum output occurs when the white dot on the sensor points towards a magnetic north. When no magnetic field is present, the sensor will read an offset voltage of about 2 volts.

A magnetic field will cause the voltage to increase or decrease depending on the direction of the field. The minimum is 0 volts, maximum is 4 volts. If the offset voltage is set incorrectly, or if the magnetic field is beyond the range of the sensor, the voltage will reach one of these limits. An amplifier used with the sensor has a switch for low and high amplification.

Low amplification is used to measure relatively strong magnetic fields around permanent magnets and electromagnets. Each volt represents $$32~ gauss(3.2\cdot 10^{-3} ~tesla)$$.

High amplification is used mainly to measure the magnetic field of the earth and very weak fields. It can be used for other magnets, but the sensor must remain in one position so that the reading is not affected by the background field of the earth. It is 20 times more sensitive than the low amplification. Each volt represents $$1.6~gauss (1.6 \cdot 10^{-4}~tesla)$$. The range of the sensor is $$\pm 3.2~gauss$$.

## Procedure

### --- 1st period, General Setup ---

 Magnetic Field Measurements Figure 4

### 1) Magnetic Field Sensor

Turn on the computer and login to the network. Run the "Internet Explorer" web browser and select "CapMagnetism" from the favorites menu. This will run software that works with the computer interface and has been setup to measure magnetic fields with the magnetic field sensor, the amplifier set to high.

For your work, you will need to scale the computer data graphs to suit your measurements. Also zero has not been set (absolute measurement can not be made, only relative measurements).

Test your sensor and software to obtain an understanding of how it functions. Notice the sensor is directional and readings are very sensitive to the probes orientation in 3-space. The magnetic field is measured perpendicular to the surface of the white dot on the sensor. Test the sensor operation in the field of a permanent magnet.

### 2) Earth's magnetic field and inclination

Set the amplifier to high and hold the probe away from possible magnetic fields and any metal. Measure earth's magnetic field by orienting the probe for a maximum reading and then reversing the probe for a maximum negative reading (software scaling will need to be adjusted). Earth's field will be half the difference. By taking data as the difference in reversed probe orientation will account for any zero offset and asymmetries in the experiment.

If the sensor tube is held vertically and rotated until the maximum reading is found, magnetic north will be perpendicular to the white sensor dot. The magnetic inclination in the area can be found by holding the tube so that the white dot is facing north and rotating the sensor end of the tube down until the reading reaches a maximum. Determine the magnetic inclination (the angle from horizontal) of the earth's field at your bench. (Your measured field strength and inclination may be strongly influenced by surroundings including in floors and walls).

## Procedure

### 1) Circuit Setup

 Figure 5

Multiple batteries will be used as a variable voltage/current source. Organize your workspace with the circuit if figure 5 so that you can easily add/remove batteries while measuring magnetic field intensities within the wire coils. Be careful not to short the batteries. Test the field of the coil, adjusting the software scaling to suit.

This lab requires fairly large currents in the wires. Do not leave the circuit on for extended times as it drains the batteries and heats the wires. Only switch the circuit on when you are making a measurement. Special caution on cleanup not to short the batteries.

### 2) Magnetic Field and Coil Current Experiment

To determine how the magnetic field at the center of a coil varies with the current through the coil.

Setup the circuit of figure 5 with a coil of as many loops as can be made with the length of wire given. Orient the magnetic field sensor at the coil center to measure along the expected field direction (the sensor measures the field perpendicular to the surface of its white dot). Tape the sensor in place so it remains in the same position for the experiment.

Zero the sensor when no current is flowing, removing the effect of earth's magnetic field and any local magnetism. Prepare to record the data in a table along with drawing a rough graph.

Record the field for a number of different currents. Check zero with each measurement (the field of interest is the difference between current and no current). Determine the slope of your rough graph.

Relate the graph slope to the theory equations. Calculate what is expected from theory for the graph slope (if not done during the period then complete this at home and insert into your notebook during the following period).

## Procedure

Figure 6

### 1) Solenoid Magnetic Field Overview

Setup the computer as in the 1st period's general setup (note, the sensor amplification needs to be set on high) and setup the circuit as in figure 5 with the Slinky as the coil. Stretch the Slinky such that the magnetic sensor will fit between the coils and tape it into place. Determine the number of turns per length for your solenoid setup.

Obtain a sense of how the relative magnetic field varies in and around the solenoid. Test the center of the Slinky. Note the relative field strength with the sensor pointing along the axis as well as perpendicular to the axis of the solenoid. Explore how the field strength varies along the solenoid length. Note how the field is distributed outside the solenoid.

Note, there are many magnetic fields (including earth's field) that are being imposed on your sensor readings. Your interest here is only in the field due to current flowing in the solenoid. You can determine this by simply turning the current on/off and noting the difference due to this change. The software also has a button that can set zero to current value.

The Slinky is also made of an iron alloy that can magnetize itself. Moving the Slinky around can change the local field even with no current flowing. Hence the sensor measurements will need to be either zeroed each time the Slinky is moved or recorded as the difference between current on and off

### 2) Magnetic Field and Solenoid Coil Spacing Experiment

The goal is to compare theory with experiment of how the magnetic field in the solenoid relates to the spacing of its turns.

• From theory, determine what would need to be plotted to result in a linear graph. Determine what the expected slope will be from theory.

Fix the sensor at the center of the Slinky and orient the sensor to measure the magnetic field along the Slinky long axis. Zero the sensor reading to remove effects from earth's and any other fields. Prepare to record the data in a table along with drawing a graph. Take some preliminary measurements to determine the extremes of your graph as well as obtaining a general sense of uncertainties involved.

• With a single current, record data (into a table and onto your graph at the same time) of the magnetic field at the solenoid center as a function of solenoid coil spacing by changing the length of the Slinky. With uncertainties, determine the slope of your graph and compare to theory.

The field certainty is how well the experiment can be done, not how exact the field value is given by the software. With the uncertainty, consider the reproducibility of your result as well as randomness in your measurements (including when no current is flowing).

Notes