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Physics LibreTexts

17: Magnetic Dipole Moment

A book says that the equivalent width W, in wavelength units, of a spectrum line, is related to the number of atoms per unit area in the line of sight, N, by

                                                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif                                                         16.6.1

Is this formula all right in any system of units? Can I use SI units on the right hand side, and get the answer in metres? Or must I use a particular set of units in order to get the right answer? And if so, which units?

A book says that the rate at which energy is radiated, P, from an accelerating charge is

                                                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif                                                             16.6.2

Is this correct? Is c the speed of light, or is it merely a conversion factor between different units? Or is one of the cs a conversion factor, and the other two are the speed of light?

It is possible to find the answer to such bewildering questions, if we do a bit of dimensional analysis. So, before trying to answer these specific questions (which I shall do later as examples) I am going to present a table of dimensions. I already gave a table of dimensions of electrical quantities in Chapter 11, in terms of M, L, T and Q, but that table won’t be particularly helpful in the present context.

I pointed out in Section 16.1 of the present chapter that Coulomb’s law is often written in the form

                                                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif                                                              16.6.3

Consequently the dimensions of Q are held to be File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif  But we know that a permittivity is missing from the denominator of equation 16.6.3, because the writer intends his formula to be restricted to a particular set of units such that k or 4pe0 = 1. In order to detect whether a permittivity has been omitted from an equation, we need a table in which the dimensions of electrical quantities are given not in terms of M, L, T and Q as in Chapter 11, but in terms of M, L, T and e, and this is what I am just about to do. However, often it is the permeability that has been omitted from an equation, and, in order to detect whether this is so, I am also supplying a table in which the dimensions of electrical quantities are given in terms of M, L, T and m.

If, from dimensional analysis, you find that an expression is dimensionally wrong by a power of the permittivity, insert 4pe0 in the appropriate part of the equation. If you find that an expression is dimensionally wrong by a power of the permeability, insert m0/(4p) in the appropriate part of the equation. If you find that the equation is wrong by LT-1, insert or delete c as appropriate. Your equation will then balance dimensionally and will be ready for use in any coherent system of units, including SI. This procedure will probably work in most cases, but I cannot guarantee that it will work in all cases, because it cannot deal with those (frequent!) cases in which the formula given is plain wrong, whatever units are used!

  M L T \(\epsilon\)   M L T \mu\)

Electric charge      

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 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif

-1

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

0

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

Electric dipole moment

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-1

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0

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Current

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 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif

-2

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-1

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

Potential difference

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 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

-1

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

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-2

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image013.gif

Resistance

0

-1

1

-1

 

0

 1

-1

1

Resistivity

0

  0

1

-1

 

0

 2

-1

1

Conductance

1

-1

1

 

0

-1

1

-1

Conductivity

0

-1

1

 

0

-2

1

-1

Capacitance

1

  0

1

 

0

-1

2

-1

Electric field E

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

-1

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-2

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image013.gif

Electric field D

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-1

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0

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

Electric flux FE

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif

-1

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image021.gif

-2

 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

Electric flux FD

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif

-1

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

0

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

Permittivity

0

0

0

1

 

0

-2

2

-1

Magnetic field B

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image019.gif

0

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

-1

 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

Magnetic field H

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

-2   

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File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

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-1

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

Magnetic flux FB

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

0

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

-1

 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

Magnetic flux FH

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 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image017.gif

-2   

 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

-1

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

Permeability

0

-2

2

 -1

 

 0

0     

1

Magnetic vector potential

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

0

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image015.gif

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

-1

 File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif

Inductance

0

-1

2

 -1

 

 0

1     

1

Now let’s look at the equation for equivalent width of a spectrum line:

                                                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif                                                         16.6.1

Here [W] = L and [N] = L-2. By making use of the table we find that the dimensions of the right hand side are Le. There is therefore a 4pe0 missing from the denominator, and the equation should be

                                                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image023.gif                                                       16.6.4

How about the rate at which energy is radiated from an accelerating charge?

                                                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif                                                             16.6.2

Power has dimensions ML2T-3, whereas the dimensions of the right hand side are ML2T-3e, so again there is a 4pe0 missing from the denominator and the formula should be

                                                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image025.gif                                                           16.6.5

It is often the case that there is a 4pe0 missing from the denominator is formulas that have an e2 upstairs.

“Electromagnetic” formulas often give more difficulty. For example, one book says that the energy per unit volume in a magnetic field in vacuo is File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image027.gif, while another says that is it is File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif Which is it (if indeed it is either)? Energy per unit volume has dimensions ML-1T-2. The dimensions of B2 are ML-1T-2m. The equation given is therefore wrong dimensionally by permeability, and the equation should be divided by m0/(4p) to give B2/(2m0), which is correct. On the other hand, the dimensions of H2 are ML-1T-2m-1, so perhaps we should multiply by m0/(4p)? But this does not give a correct answer, and it exemplifies some of the many difficulties that are caused by writing formulas that do not balance dimensionally and are intended to be used only with a particular set of units. The situation is particularly difficult with respect to magnetic moment, a subject to which I shall devote the next chapter.