Devon1989
Jan11-11, 12:45 PM
1. The problem statement, all variables and given/known data
Prove the formula for energy of E.M field:
\varepsilon = \frac{2}{{{{\left( {2\pi } \right)}^3}}}\int {{{\vec a}^*}\left( {\vec k} \right) \cdot \vec a\left( {\vec k} \right)} {d^3}k
2. Relevant equations
\vec E\left( {\vec x} \right) = \frac{1}{{{{\left( {2\pi } \right)}^3}}}\int {{d^3}k[{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c]}
\vec B\left( {\vec x} \right) = \frac{1}{{{{\left( {2\pi } \right)}^3}}}\int {{d^3}k\frac{{\vec k}}{k} \times [{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c]}
\varepsilon = \frac{1}{2}\int {{d^3}x} \left( {\vec E{{\left( {\vec x} \right)}^2} + \vec B{{\left( {\vec x} \right)}^2}} \right)
3. The attempt at a solution
I've tried with the formula:
\varepsilon = \frac{1}{{2{{\left( {2\pi } \right)}^6}}}\int {{d^3}x} [\int {{d^3}k[{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c] \cdot \int {{d^3}k'[{e^{i\vec k' \cdot \vec x - ick't}}\vec a\left( {\vec k'} \right) + c.c]} } +
+ \int {{d^3}k\frac{{\vec k}}{k} \times [{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c] \cdot \int {{d^3}k'\frac{{\vec k'}}{{k'}} \times [{e^{i\vec k' \cdot \vec x - ick't}}\vec a\left( {\vec k'} \right) + c.c]} }
And then I tried to integrate over x first, but I didn't get anything reasonable. I would be grateful if anyone could give me a hint.
Prove the formula for energy of E.M field:
\varepsilon = \frac{2}{{{{\left( {2\pi } \right)}^3}}}\int {{{\vec a}^*}\left( {\vec k} \right) \cdot \vec a\left( {\vec k} \right)} {d^3}k
2. Relevant equations
\vec E\left( {\vec x} \right) = \frac{1}{{{{\left( {2\pi } \right)}^3}}}\int {{d^3}k[{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c]}
\vec B\left( {\vec x} \right) = \frac{1}{{{{\left( {2\pi } \right)}^3}}}\int {{d^3}k\frac{{\vec k}}{k} \times [{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c]}
\varepsilon = \frac{1}{2}\int {{d^3}x} \left( {\vec E{{\left( {\vec x} \right)}^2} + \vec B{{\left( {\vec x} \right)}^2}} \right)
3. The attempt at a solution
I've tried with the formula:
\varepsilon = \frac{1}{{2{{\left( {2\pi } \right)}^6}}}\int {{d^3}x} [\int {{d^3}k[{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c] \cdot \int {{d^3}k'[{e^{i\vec k' \cdot \vec x - ick't}}\vec a\left( {\vec k'} \right) + c.c]} } +
+ \int {{d^3}k\frac{{\vec k}}{k} \times [{e^{i\vec k \cdot \vec x - ickt}}\vec a\left( {\vec k} \right) + c.c] \cdot \int {{d^3}k'\frac{{\vec k'}}{{k'}} \times [{e^{i\vec k' \cdot \vec x - ick't}}\vec a\left( {\vec k'} \right) + c.c]} }
And then I tried to integrate over x first, but I didn't get anything reasonable. I would be grateful if anyone could give me a hint.