Edit to “Statement”
en/8.2.17.md
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| ### Statement | |||
| − | $8.2.17.$ A metal ball of radius $r$ is located in a medium with a small conductivity $\lambda$. Determine the current flowing from the ball if its potential is $V$. If such a ball is connected by an insulated wire to a lightning rod, what will be the grounding resistance? | ||
| + | $8.2.17^*.$ A metal ball of radius $r$ is located in a medium with a small conductivity $\lambda$. Determine the current flowing from the ball if its potential is $V$. If such a ball is connected by an insulated wire to a lightning rod, what will be the grounding resistance? | ||
| ### Solution | |||
| Current will flow from the metal ball due to charge leakage into the conducting medium. The current density on the surface of the ball is: | |||
| $$j = \lambda E$$ | |||
| where $E$ is the electric field strength on the surface of the ball. | |||
| The potential of the ball is related to the electric field strength on its surface: | |||
| $$E = \frac{V}{r}$$ | |||
| Then: | |||
| $$j = \lambda \frac{V}{r}$$ | |||
| The current flowing from the surface of the ball is: | |||
| $$I = j \cdot S = \lambda \frac{V}{r} \cdot 4\pi r^2 = 4\pi\lambda rV$$ | |||
| If the ball is connected by an insulated wire to a lightning rod (which has the potential of the Earth), then the grounding resistance is: | |||
| $$R = \frac{V}{I} = \frac{V}{4\pi\lambda rV} = \frac{1}{4\pi\lambda r}$$ | |||
| #### Answer | |||
| $$I = 4\pi r\lambda V$$ | |||
| $$R = \frac{1}{4\pi r\lambda}$$ | |||
| unchanged lines 27 | |||
| @@ -1,6 +1,6 @@ | |||
| ### Statement | ### Statement | ||
| $8.2.17.$ A metal ball of radius $r$ is located in a medium with a small conductivity $\lambda$. Determine the current flowing from the ball if its potential is $V$. If such a ball is connected by an insulated wire to a lightning rod, what will be the grounding resistance? | $8.2.17^*.$ A metal ball of radius $r$ is located in a medium with a small conductivity $\lambda$. Determine the current flowing from the ball if its potential is $V$. If such a ball is connected by an insulated wire to a lightning rod, what will be the grounding resistance? | ||
| ### Solution | ### Solution | ||
| Current will flow from the metal ball due to charge leakage into the conducting medium. The current density on the surface of the ball is: | Current will flow from the metal ball due to charge leakage into the conducting medium. The current density on the surface of the ball is: | ||
| $$j = \lambda E$$ | $$j = \lambda E$$ | ||
| where $E$ is the electric field strength on the surface of the ball. | where $E$ is the electric field strength on the surface of the ball. | ||
| The potential of the ball is related to the electric field strength on its surface: | The potential of the ball is related to the electric field strength on its surface: | ||
| $$E = \frac{V}{r}$$ | $$E = \frac{V}{r}$$ | ||
| Then: | Then: | ||
| $$j = \lambda \frac{V}{r}$$ | $$j = \lambda \frac{V}{r}$$ | ||
| The current flowing from the surface of the ball is: | The current flowing from the surface of the ball is: | ||
| $$I = j \cdot S = \lambda \frac{V}{r} \cdot 4\pi r^2 = 4\pi\lambda rV$$ | $$I = j \cdot S = \lambda \frac{V}{r} \cdot 4\pi r^2 = 4\pi\lambda rV$$ | ||
| If the ball is connected by an insulated wire to a lightning rod (which has the potential of the Earth), then the grounding resistance is: | If the ball is connected by an insulated wire to a lightning rod (which has the potential of the Earth), then the grounding resistance is: | ||
| $$R = \frac{V}{I} = \frac{V}{4\pi\lambda rV} = \frac{1}{4\pi\lambda r}$$ | $$R = \frac{V}{I} = \frac{V}{4\pi\lambda rV} = \frac{1}{4\pi\lambda r}$$ | ||
| #### Answer | #### Answer | ||
| $$I = 4\pi r\lambda V$$ | $$I = 4\pi r\lambda V$$ | ||
| $$R = \frac{1}{4\pi r\lambda}$$ | $$R = \frac{1}{4\pi r\lambda}$$ | ||
| unchanged lines 27 | |||