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en/6.5.23.md
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| + | ### Statement | ||
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| + | $6.5.23.$ [Insert the problem statement] | ||
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| + | ### Solution | ||
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| + | Suppose the potential energy of an isolated face of the tetrahedron is $U_1$. Then, the potential energy of two faces of the tetrahedron pressed together is $4U_1$, because the charge is doubled in the same space. Let the potential energy of two faces of the tetrahedron making a dihedral angle with each other be $2U_1+U_2$. Then the work done in pressing the two faces together is $A=4U_1-(2U_1+U_2)=2U_1-U_2$. | ||
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| + | Originally, the potential energy of the tetrahedron is $4U_1+6U_2$, because there are $6$ possible pairs out of $4$ faces. When the four faces are pressed together, the potential energy is $16U_1$, because the charge is quadrupled in the same space. Thus, the work done in collapsing the tetrahedron is $16U_1-(4U_1+6U_2)=12U_1-6U_2=6A$. | ||
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| + | #### Answer | ||
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| + | [Insert a concise answer or boxed result] | ||
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| ### Statement | |||
| $6.5.23.$ [Insert the problem statement] | |||
| ### Solution | |||
| Suppose the potential energy of an isolated face of the tetrahedron is $U_1$. Then, the potential energy of two faces of the tetrahedron pressed together is $4U_1$, because the charge is doubled in the same space. Let the potential energy of two faces of the tetrahedron making a dihedral angle with each other be $2U_1+U_2$. Then the work done in pressing the two faces together is $A=4U_1-(2U_1+U_2)=2U_1-U_2$. | |||
| Originally, the potential energy of the tetrahedron is $4U_1+6U_2$, because there are $6$ possible pairs out of $4$ faces. When the four faces are pressed together, the potential energy is $16U_1$, because the charge is quadrupled in the same space. Thus, the work done in collapsing the tetrahedron is $16U_1-(4U_1+6U_2)=12U_1-6U_2=6A$. | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||