- #1
- 1,752
- 143
Help with units
The book gives the following forumla
[tex]E_0 = \sqrt {2\mu _0 c\left\langle S \right\rangle } [/tex]
And when it uses it in an example, the units go from:
[tex]\sqrt {\left( {{\rm{H/m}}} \right)\left( {{\rm{m/s}}} \right)\left( {{\rm{W/m}}^{\rm{2}} } \right)} [/tex]
to [tex]V/m[/tex], with no intermediate steps. My two efforts fall short:
Attempt #1
[tex]\sqrt {\left( {{\rm{H/m}}} \right)\left( {{\rm{m/s}}} \right)\left( {{\rm{W/m}}^{\rm{2}} } \right)} = \sqrt {\frac{{{\rm{HmW}}}}{{{\rm{msmm}}}}} = \sqrt {\frac{{{\rm{HW}}}}{{{\rm{m}}^{\rm{2}} {\rm{s}}}}} [/tex]
Does a Henry - Watt divided by a second equal a Volt squared? If so, = V/m
Attempt #2 using N/A^2 instead of H/m for units of mu0
[tex]\begin{array}{l}
\sqrt {\left( {{\rm{N/A}}^{\rm{2}} } \right)\left( {{\rm{/s}}} \right)\left( {{\rm{W/}}{\rm{m}}} \right)} = \\
\\
\sqrt {\frac{{{\rm{NW}}}}{{{\rm{A}}^{\rm{2}} {\rm{s}} \cdot {\rm{m}}}}} = \sqrt {\frac{{{\rm{kgmJ}}}}{{{\rm{A}}^{\rm{2}} {\rm{s}} \cdot {\rm{ms}}^{\rm{2}} {\rm{s}}}}} = \sqrt {\frac{{{\rm{kg - m - kg - m - m}}}}{{\left( {\frac{{\rm{C}}}{{\rm{s}}}} \right)^{\rm{2}} {\rm{s - m - s}}^{\rm{2}} {\rm{s - s}}^{\rm{2}} }}} = \sqrt {\frac{{{\rm{kg - m - kg - m - }}{\rm{ - }}}}{{{\rm{C}}^{\rm{2}} {\rm{s - }}{\rm{ - s}}^{\rm{2}} {\rm{ - s - }}}}} \\
\\
= \sqrt {\frac{{{\rm{kg - m - kg - m}}}}{{{\rm{C}}^{\rm{2}} {\rm{s - s}}^{\rm{2}} {\rm{ - s}}}}} = \sqrt {\frac{{{\rm{kg - m - kg - m}}}}{{\left( {{\rm{FV}}} \right)^{\rm{2}} {\rm{s - s}}^{\rm{2}} {\rm{ - s}}}}} = \sqrt {\frac{{{\rm{kg - m - kg - m}}}}{{{\rm{F}}^{\rm{2}} {\rm{V}}^{\rm{2}} {\rm{ - s - s}}^{\rm{2}} {\rm{ - s}}}}} = \sqrt {\frac{{{\rm{NW}}}}{{{\rm{A}}^{\rm{2}} {\rm{s - m}}}}} = \sqrt {\frac{{{\rm{kg - }}{\rm{ - mJ}}}}{{{\rm{A}}^{\rm{2}} {\rm{s}}{\rm{ - s - s}}^{\rm{2}} }}} = \\
\\
\sqrt {\frac{{{\rm{kg - mJ}}}}{{{\rm{A}}^{\rm{2}} {\rm{s - s - s}}^{\rm{2}} }}} \\
\end{array}[/tex]
This is not beginning to resemble [tex]\sqrt {\frac{{{\rm{V}}^{\rm{2}} }}{{{\rm{m}}^2 }}} [/tex]
The book gives the following forumla
[tex]E_0 = \sqrt {2\mu _0 c\left\langle S \right\rangle } [/tex]
And when it uses it in an example, the units go from:
[tex]\sqrt {\left( {{\rm{H/m}}} \right)\left( {{\rm{m/s}}} \right)\left( {{\rm{W/m}}^{\rm{2}} } \right)} [/tex]
to [tex]V/m[/tex], with no intermediate steps. My two efforts fall short:
Attempt #1
[tex]\sqrt {\left( {{\rm{H/m}}} \right)\left( {{\rm{m/s}}} \right)\left( {{\rm{W/m}}^{\rm{2}} } \right)} = \sqrt {\frac{{{\rm{HmW}}}}{{{\rm{msmm}}}}} = \sqrt {\frac{{{\rm{HW}}}}{{{\rm{m}}^{\rm{2}} {\rm{s}}}}} [/tex]
Does a Henry - Watt divided by a second equal a Volt squared? If so, = V/m
Attempt #2 using N/A^2 instead of H/m for units of mu0
[tex]\begin{array}{l}
\sqrt {\left( {{\rm{N/A}}^{\rm{2}} } \right)\left( {{\rm{/s}}} \right)\left( {{\rm{W/}}{\rm{m}}} \right)} = \\
\\
\sqrt {\frac{{{\rm{NW}}}}{{{\rm{A}}^{\rm{2}} {\rm{s}} \cdot {\rm{m}}}}} = \sqrt {\frac{{{\rm{kgmJ}}}}{{{\rm{A}}^{\rm{2}} {\rm{s}} \cdot {\rm{ms}}^{\rm{2}} {\rm{s}}}}} = \sqrt {\frac{{{\rm{kg - m - kg - m - m}}}}{{\left( {\frac{{\rm{C}}}{{\rm{s}}}} \right)^{\rm{2}} {\rm{s - m - s}}^{\rm{2}} {\rm{s - s}}^{\rm{2}} }}} = \sqrt {\frac{{{\rm{kg - m - kg - m - }}{\rm{ - }}}}{{{\rm{C}}^{\rm{2}} {\rm{s - }}{\rm{ - s}}^{\rm{2}} {\rm{ - s - }}}}} \\
\\
= \sqrt {\frac{{{\rm{kg - m - kg - m}}}}{{{\rm{C}}^{\rm{2}} {\rm{s - s}}^{\rm{2}} {\rm{ - s}}}}} = \sqrt {\frac{{{\rm{kg - m - kg - m}}}}{{\left( {{\rm{FV}}} \right)^{\rm{2}} {\rm{s - s}}^{\rm{2}} {\rm{ - s}}}}} = \sqrt {\frac{{{\rm{kg - m - kg - m}}}}{{{\rm{F}}^{\rm{2}} {\rm{V}}^{\rm{2}} {\rm{ - s - s}}^{\rm{2}} {\rm{ - s}}}}} = \sqrt {\frac{{{\rm{NW}}}}{{{\rm{A}}^{\rm{2}} {\rm{s - m}}}}} = \sqrt {\frac{{{\rm{kg - }}{\rm{ - mJ}}}}{{{\rm{A}}^{\rm{2}} {\rm{s}}{\rm{ - s - s}}^{\rm{2}} }}} = \\
\\
\sqrt {\frac{{{\rm{kg - mJ}}}}{{{\rm{A}}^{\rm{2}} {\rm{s - s - s}}^{\rm{2}} }}} \\
\end{array}[/tex]
This is not beginning to resemble [tex]\sqrt {\frac{{{\rm{V}}^{\rm{2}} }}{{{\rm{m}}^2 }}} [/tex]
Last edited by a moderator: