### Purpose

To measure and determine the relationship between a magnetic field generated by a line of current and a radial distance from a conductor, and to measure and determine the relationship between a magnetic field at the center of a coil and the number of turns in a coil.

### Hypothesis

As the distance from the center of the conductor increases, the magnetic field strength will decrease in accordance with the equation B = μ_{o}I / 2πR. As the number of turns in the coil increases, the magnetic field strength will increase, in accordance with the equation B = Nμ_{o}I / 2R.

### Labeled Diagrams

See attached sheet.

### Data

*Part 1*

Current = 7 A

R (m) |
B (mT) |
B (T) |
1/R (m |

0.0050 |
0.17 |
0.00017 |
200.0 |

0.010 |
0.12 |
0.00012 |
100.0 |

0.015 |
0.074 |
0.000074 |
66.66 |

0.020 |
0.061 |
0.000061 |
50.00 |

0.025 |
0.044 |
0.000044 |
40.00 |

0.030 |
0.041 |
0.000041 |
33.33 |

0.035 |
0.036 |
0.000036 |
28.57 |

0.040 |
0.031 |
0.000031 |
25.00 |

0.045 |
0.027 |
0.000027 |
22.22 |

0.050 |
0.023 |
0.000023 |
20.00 |

0.055 |
0.019 |
0.000019 |
18.18 |

0.060 |
0.018 |
0.000018 |
16.66 |

0.065 |
0.015 |
0.000015 |
15.38 |

0.070 |
0.013 |
0.000013 |
14.28 |

0.075 |
0.011 |
0.000011 |
13.33 |

0.080 |
0.010 |
0.000010 |
12.50 |

0.085 |
0.0080 |
0.0000080 |
11.76 |

0.090 |
0.0050 |
0.0000050 |
11.11 |

0.095 |
0.0030 |
0.0000030 |
10.52 |

0.10 |
0.0020 |
0.0000020 |
10.00 |

*Part 2*

Current = 7 A

Diameter of Cylindrical support = 0.130 m

# of turns N |
B (mT) |
B (T) |

1 |
0.0880 |
0.0000880 |

2 |
0.171 |
0.000171 |

3 |
0.242 |
0.000242 |

4 |
0.312 |
0.000312 |

5 |
0.371 |
0.000371 |

6 |
0.411 |
0.000411 |

### Graphs

*Part 1*

*Part 2*

### Questions

*Part 1*

1. Theory states that the magnetic field produced by a long straight current-carrying wire decreases in strength as you get further from the wire. The exact dependence of the magnetic field strength *B* on radial distance from the wire R is B = μ_{o}I / 2πR where μ_{o} is the permeability of free space and has a value of 4π x 10^{-7} Tm/A. For your data from Part 1, plot a graph in Graphical Analysis of *B* vs. 1/R. Your graph should have a linear trend. Perform a Linear Fit on your graph. What is the value of the slope?

The slope is 9.2 x 10^{-7} Tm.

2. Calculate your value of μ_{o}I / 2π. How does it compare to the slope of your graph? What is the percent difference?

The value is 1.4 x10^{-6} Tm, which is larger in value in comparison to the slope. The percent difference is 41%.

*Part 2*

3. Theory states that the magnetic field produced by a circular loop of current- carrying wire increases in strength as the number of turns of wire is increased. The exact dependence of the magnetic field strength *B* on the number of turns *N* is B = Nμ_{o}I/ 2R where R is the radius of the loop of wire. For your data from Part 2, plot a graph in Graphical Analysis of *B* vs. *N*. Your graph should have a linear trend. Perform a Linear Fit on your graph. What is the value of the slope?

The graph of the slope is 6.53 x 10^{-5} T/turn.

4. Calculate your value of μ_{o}I / 2R. How does it compare to the slope of your graph? What is the percent difference?

The value is 6.7 x 10^{-5} T/turn, which is very similar yet slightly large in value than the slope. The percent difference is 2.6%.

### Conclusion

During part one of the experiment, magnetic field strength was measured as a function of radial distance from a conductor. First, a piece of polar graph paper with concentric circles starting at a diameter of 0.5 cm increasing in increments of 0.5 cm to 10 cm was punched through a rigid aluminum conductor at the center of the concentric circles. The paper was placed on the plastic table of the apparatus and was aligned using a compass so that the parallel lines on the sheet were pointing north. The paper was then secured to the apparatus using tape. The high amperage DC power supply was connected in series with a high power resistor and the aluminum wire at the side and on top of the apparatus. The magnetic field sensor was zeroed and the DC power supply was set to 7 A. With the current on and kept constant, the magnetic field strength was recorded at each circle on the polar graph paper by holding the sensor in line with the parallel lines on the sheet and so that the white dot on the sensor was on the left and at a 90^{o} angle with the parallel lines pointing north. The magnetic field strength was recorded using Vernier Lab Pro. This value was recorded along with the respective radius.

Graphical Analysis was used to plot B vs. 1/R and perform a linear trend. The resulting slope was 9.2 x 10^{-7} Tm and the correlation was 0.9734. It was somewhat difficult to get accurate readings at the smaller radii, which negatively affected this correlation. The slope from this plot in comparison to the value of μ_{o}I / 2π, 1.4 x10^{-6} Tm, yielded a percent difference of 41%. The data followed the expected trend of decreasing in magnetic field strength as the radius increased. Possible sources of error include the difficulty of aligning the sensor perfectly along the radius of each circle, and it was also a challenge to get an accurate reading from the sensor as the readings kept jumping around. Additionally, if the sheet was not perfectly aligned northward, there would be interference from the earth’s magnetic pull which would have affected the readings.

During part two of the experiment, magnetic field strength was measured as a function of the number of turns in a wire. First, the power supply was turned off and the center aluminum wire was removed along with the top table of the tangential galvanometer. The magnetic field sensor was inserted into the side of the cylindrical support by screwing it into the side of the top table support using a threaded plastic bushing. The white dot of the sensor was aligned to be in the center of the cylindrical support with the white dot facing upward. The sensor was zeroed and the power supply was turn on to 7 A. A piece of flexible wire was wound around the outside of this support to complete one full turn. The magnetic field sensor measured B using “Events with Entry” on Lab Pro. The magnetic field strength was measured for six full turns of the wire. The diameter of the cylindrical support was also measured, being 0.013 m.

Graphical Analysis was used to plot B vs N and perform a linear trend. The resulting slope was 6.53 x 10^{-5} T/turn and the correlation was 0.9943. The data followed the expected trend of increasing in magnetic field strength as the number of turns increased. The slope from this plot in comparison to the value of μ_{o}I / 2R, 6.7 x 10^{-5} T/turn, yielded a percent difference of only 2.6%. Possible error could have resulted from the earth’s magnetic field. Additionally, the wire was not perfectly wrapped around the base, which would throw the readings off as it would not have been an exact turn of the wire. Finally, it was difficult to align the sensor perfectly in the middle of the support, so again this would contribute to some error in the readings.

### Equations

Percent Difference = |E2 – E1| / [ (E1 + E2) / 2 ] x 100%

B x R = μ_{o}I / 2π

B / N = μ_{o}I / 2R