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| <title>Two infinite plates of thickness h are charged uniformly in volume and stacked together. The bulk charge density of the first plate is \rho, and the second plate -\rho. Find the maximum electric field intensity.</title> | | <title>Two infinite plates of thickness h are charged uniformly in volume and stacked together. The bulk charge density of the first plate is \rho, and the second plate -\rho. Find the maximum electric field intensity.</title> |
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| <h3 id="back-link"><a href="../../#6.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#6.2">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $6.2.10.$ Two infinite planes intersecting at an angle $\alpha$ divide space into four regions. What is the electric field strength in regions 1 and 2 if the surface charge density of the planes is $\pm \sigma$? | | $6.2.10.$ Two infinite planes intersecting at an angle $\alpha$ divide space into four regions. What is the electric field strength in regions 1 and 2 if the surface charge density of the planes is $\pm \sigma$? |
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| <img src="statement.png" | | <img src="statement.png" |
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| <figcaption> | | <figcaption> |
| For problem $6.2.10$ | | For problem $6.2.10$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Consider the following figure... | | Consider the following figure... |
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| Field Analysis | | Field Analysis |
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| It's known that the electric field intensity for one of the faces of a infinite plane with surface charge density is $E = \frac{\sigma}{2\varepsilon_0}$. | | It's known that the electric field intensity for one of the faces of a infinite plane with surface charge density is $E = \frac{\sigma}{2\varepsilon_0}$. |
| </p> | | </p> |
| <p> | | <p> |
| For region 1 (see part $a$ of above figure): | | For region 1 (see part $a$ of above figure): |
| </p> | | </p> |
| <p> | | <p> |
| $$E_{R1y} = E - E \cos{\alpha} = \frac{\sigma}{2\varepsilon_0}(1-\cos{\alpha})$$ | | $$E_{R1y} = E - E \cos{\alpha} = \frac{\sigma}{2\varepsilon_0}(1-\cos{\alpha})$$ |
| </p> | | </p> |
| <p> | | <p> |
| and for $x$-direction, $$E_{R1x} = E \sin{\alpha} = \frac{\sigma}{2\varepsilon_0}\sin{\alpha}.$$ So, as $$E_{R1} = \sqrt{E_{R1x}^2+E_{R1y}^2},$$ and taking in account that $$\sin{\alpha} = \sqrt{\frac{1-\cos{\alpha}}{2}},$$ it is obtained | | and for $x$-direction, $$E_{R1x} = E \sin{\alpha} = \frac{\sigma}{2\varepsilon_0}\sin{\alpha}.$$ So, as $$E_{R1} = \sqrt{E_{R1x}^2+E_{R1y}^2},$$ and taking in account that $$\sin{\alpha} = \sqrt{\frac{1-\cos{\alpha}}{2}},$$ it is obtained |
| </p> | | </p> |
| <h4>Answer 1</h4> | | <h4>Answer 1</h4> |
| <p> | | <p> |
| $$E_{R1} = \frac{\sigma}{\varepsilon_0}\sin{\frac{\alpha}{2}}$$ | | $$E_{R1} = \frac{\sigma}{\varepsilon_0}\sin{\frac{\alpha}{2}}$$ |
| </p> | | </p> |
| <p> | | <p> |
| For region 2, for $y$-direction: $$E_{R2y} = E(1+\cos{\alpha})=\frac{\sigma}{2\varepsilon_0}(1+\cos{\alpha})$$ and for $x$-axis, $$E_{R2x} = E\sin{\alpha} = \frac{\sigma}{2\varepsilon_0}\sin{\alpha}.$$ | | For region 2, for $y$-direction: $$E_{R2y} = E(1+\cos{\alpha})=\frac{\sigma}{2\varepsilon_0}(1+\cos{\alpha})$$ and for $x$-axis, $$E_{R2x} = E\sin{\alpha} = \frac{\sigma}{2\varepsilon_0}\sin{\alpha}.$$ |
| </p> | | </p> |
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| Again, $$E_{R2} = \sqrt{E_{R2x}^2+E_{R2y}^2}$$ and taking in account that $$\cos{\frac{\alpha}{2}} = \sqrt{\frac{1+\cos{\alpha}}{2}}$$ | | Again, $$E_{R2} = \sqrt{E_{R2x}^2+E_{R2y}^2}$$ and taking in account that $$\cos{\frac{\alpha}{2}} = \sqrt{\frac{1+\cos{\alpha}}{2}}$$ |
| </p> | | </p> |
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| <h4>Answer 2</h4> | | <h4>Answer 2</h4> |
| <p> | | <p> |
| $$E_{R2} = \frac{\sigma}{\varepsilon_0}\cos{\frac{\alpha}{2}}$$ | | $$E_{R2} = \frac{\sigma}{\varepsilon_0}\cos{\frac{\alpha}{2}}$$ |
| </p> | | </p> |
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| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<br> |
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