# Physics307L:People/Dougherty/Notebook/071008

## Contents

## Cary M. Dougherty 15:34, 8 October 2007 (CDT)

## Setup/Prodedure

^{SJK 00:52, 13 November 2007 (CST)}The heating element connected to the anode gun is heated to the point of boiling off electrons which are then accelerated by a voltage across a circular plate. the electrons in motion then move in a "curled" path by a magnetic field caused by a current in the helmholtz coils. this magnetic field caused by the current constantly puts a force on the electrons making them turn in a circular "orbit." then the helium in the bulb causes collisions with the electrons causing a bluish radiation of visible light. we can measure the radius of the beam by a ruler behind the bulb. by measuring the voltage applied, the heat applied, and the current, along with the radius of the beam, we can come up with the e/m ratio.

## same voltage, different current 1

Accelerating Voltage | Coil Current | Left Radius (cm) +/-0.1 | Right Radius (cm) +/-0.1 | Avg Radius (cm) +/- |
---|---|---|---|---|

246 | 1.02 | 4.5 | 4.9 | 4.7 |

246 | 1.04 | 4.4 | 4.8 | 4.6 |

246 | 1.06 | 4.4 | 4.6 | 4.5 |

246 | 1.08 | 4.4 | 4.6 | 4.5 |

246 | 1.10 | 4.4 | 4.6 | 4.5 |

246 | 1.12 | 4.4 | 4.5 | 4.45 |

246 | 1.14 | 4.4 | 4.5 | 4.45 |

246 | 1.16 | 4.4 | 4.5 | 4.45 |

246 | 1.18 | 4.4 | 4.4 | 4.4 |

246 | 1.20 | 4.4 | 4.3 | 4.35 |

246 | 1.22 | 4.3 | 4.2 | 4.25 |

246 | 1.24 | 4.2 | 4.2 | 4.2 |

246 | 1.26 | 4.2 | 4.0 | 4.1 |

## same voltage, different current 2

Accelerating Voltage (V) | Coil Current (A) | Left Radius (cm) +/-0.1 | Right Radius (cm) +/-0.1 | Avg. Radius (cm) +/- 0. |
---|---|---|---|---|

246 | 1.02 | 4.5 | 4.6 | 4.55 |

246 | 1.04 | 4.5 | 4.6 | 4.55 |

246 | 1.06 | 4.5 | 4.5 | 4.5 |

246 | 1.08 | 4.5 | 4.5 | 4.5 |

246 | 1.10 | 4.5 | 4.5 | 4.5 |

246 | 1.12 | 4.5 | 4.5 | 4.5 |

246 | 1.14 | 4.5 | 4.4 | 4.45 |

246 | 1.16 | 4.4 | 4.4 | 4.4 |

246 | 1.18 | 4.4 | 4.3 | 4.35 |

246 | 1.20 | 4.4 | 4.3 | 4.35 |

246 | 1.22 | 4.3 | 4.3 | 4.3 |

246 | 1.24 | 4.3 | 4.2 | 4.25 |

246 | 1.26 | 4.2 | 4.1 | 4.15 |

## change heater

Changed heater voltage to 6.0 from 6.2

Accelerating Voltage (V) | Coil Current (A) | Left Radius (cm) +/-0.1 | Right Radius (cm) +/-0.1 | Avg. Radius (cm) +/- |
---|---|---|---|---|

246 | 1.02 | 3.9 | 4.0 | 3.95 |

246 | 1.04 | 3.9 | 4.0 | 3.95 |

246 | 1.06 | 3.9 | 3.9 | 3.9 |

246 | 1.08 | 3.9 | 3.9 | 3.9 |

246 | 1.10 | 3.9 | 3.9 | 3.9 |

246 | 1.12 | 3.9 | 3.9 | 3.9 |

246 | 1.14 | 3.9 | 3.9 | 3.9 |

## same current, different voltage

Accelerating Voltage (V) | Wire Current (A) | Left Radius (cm) +/- .1 | Right Radius (cm) +/- .1 | Avg Radius | Radius SqauredAvg |
---|---|---|---|---|---|

250 | 1.31 | 4.3 | 4.0 | 4.15 | 17.2225 |

255 | 1.31 | 4.3 | 4.1 | 4.2 | 17.64 |

260 | 1.31 | 4.4 | 4.1 | 4.26 | 18.0625 |

265 | 1.31 | 4.5 | 4.2 | 4.35 | 19.9225 |

270 | 1.31 | 4.5 | 4.2 | 4.35 | 19.9225 |

275 | 1.31 | 4.5 | 4.3 | 4.4 | 19.36 |

280 | 1.31 | 4.5 | 4.3 | 4.4 | 19.36 |

285 | 1.31 | 4.5 | 4.4 | 4.45 | 19.8025 |

290 | 1.31 | 4.6 | 4.4 | 4.5 | 20.25 |

295 | 1.31 | 4.6 | 4.5 | 4.55 | 20.7025 |

300 | 1.31 | 4.6 | 4.5 | 4.55 | 20.7025 |

## Calculating e/m

To calculate e/m ratio we can use a sum of forces (ignoring gravity) and with Newton's 2nd law and centripetal motion we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle F=ma}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle evB=\frac{mv^2}{r}}**

solve for q/m using:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle qV=\frac{1}{2}mv^2}**

and solve for v and plugging in:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle e/m=\frac{2V}{(B*I)^2R^2}}**

where B=7.8 x 10^-4 (weber/amp-meter^2)

## e/m calculations

^{SJK 00:54, 13 November 2007 (CST)}#### Set 1

^{SJK 00:56, 13 November 2007 (CST)}V=[246,246,246,246,246,246,246,246,246,246,246,246,246];

I=[1.02,1.04,1.06,1.08,1.10,1.12,1.14,1.16,1.18,1.20,1.22,1.24,1.26];

r=[4.7,4.6,4.5,4.5,4.5,4.45,4.45,4.45,4.4,4.35,4.25,4.2,4.1];

B=7.8e-4.*I;

e_m=(2.*V)./((B.^2).*(r.^2))

avg=mean(e_m)

e_m =

1.0e+011 *

Columns 1 through 5

3.5187 3.5334 3.5542 3.4238 3.3004

Columns 6 through 10

3.2555 3.1423 3.0349 2.9999 2.9678

Columns 11 through 13

3.0080 2.9815 3.0302

avg =

3.2116e+011

#### Set 2

close all;clear all;

V=[246,246,246,246,246,246,246,246,246,246,246,246,246];

I=[1.02,1.04,1.06,1.08,1.10,1.12,1.14,1.16,1.18,1.20,1.22,1.24,1.26];

r=[4.55,4.55,4.5,4.5,4.5,4.5,4.45,4.4,4.35,4.35,4.3,4.25,4.15];

B=7.8e-4.*I;

e_m=(2.*V)./((B.^2).*(r.^2))

avg=mean(e_m)

e_m =

1.0e+011 *

Columns 1 through 5

3.7545 3.6115 3.5542 3.4238 3.3004

Columns 6 through 10

3.1836 3.1423 3.1042 3.0693 2.9678

Columns 11 through 13

2.9385 2.9118 2.9576

avg =

3.2246e+011

#### Set 3

close all; clear all;

V=[246,246,246,246,246,246,246];

I=[1.02,1.04,1.06,1.08,1.10,1.12,1.14];

r=[3.95,3.95,3.9,3.9,3.9,3.9,3.9];

B=7.8e-4.*I;

e_m=(2.*V)./((B.^2).*(r.^2))

avg=mean(e_m)

e_m =

1.0e+011 *

Columns 1 through 5

4.9817 4.7920 4.7319 4.5583 4.3940

Columns 6 through 7

4.2385 4.0911

avg =

4.5411e+011

#### Set 4

close all;clear all;

V=[250,255,260,265,270,275,280,285,290,295,300];

I=[1.31,1.31,1.31,1.31,1.31,1.31,1.31,1.31,1.31,1.31,1.31];

r=[4.15,4.2,4.26,4.35,4.35,4.4,4.4,4.45,4.5,4.55,4.55];

B=7.8e-4.*I;

e_m=(2.*V)./((B.^2).*(r.^2))

e_m =

1.0e+011 *

Columns 1 through 5

2.7806 2.7691 2.7444 2.6827 2.7333

Columns 6 through 10

2.7210 2.7705 2.7569 2.7433 2.7296

Column 11

2.7759

avg =

2.7461e+011

## Standard error of the mean

Standard Error :**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2} }**

**SET 1**: s=2.3052e+010

**SET 2**: s=2.8116e+010

**SET 3**: s=3.1873e+010

**SET 4**: s=2.9017e+009

Standard error of the mean: **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle SE = \frac{s}{\sqrt{N}} }**

**SET 1**: SE= 6.3934e+009

**SET 2**: SE= 7.7978e+009

**SET 3**: SE= 1.2047e+010

**SET 4**: SE= 8.7489e+008

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle %error= \frac{|Actual-Experimental|}{|Actual|}x100}**

**SET 1**: e= 82%

**SET 2**: e= 83%

**SET 3**: e= 158%

**SET 4**: e= 56%

## Possibilities of errors

The lab manual discusses why the e/m ratio we came up with is higher than normal. it says that the voltage across the anode creates an un-uniform field which causes the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_e}**
to slow down and also the collisions with the helium and electrons also causes **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_e}**
to slow down. and since that is directly related to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1/(R^2)}**
, then it greatly effects the radius measured.

another few problems is the width of the beam, although the manual says the best way to measure is the outside of the beam. also it is hard to line up the beam with the ruler behind the bulb because of light and the length measured (mm). another couple problems is the bulb itself. the circular bulb creates a parallax of the light viewed and can distort your measurements. and also the bulb itself is very unstable. lining it up so its center is directly over the origin of the ruler is tough and also if the table is bumped or the apparatus is bumped it can move the bulb and skew measurements.

as of the voltmeter and ampmeter. as discussed in class. they are directly hooked up and directly measured so the probability of error is surely less then 1%.