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A vacuum diode is shown at right. The heated cathode emits electrons into the region between two parallel plates, where they travel towards the anode. For a fixed anode voltage Vo, the system reaches a steady-state described by a static charge distribution p(x). The plates are separated by a distance a, and the electrostatic potential is cathode Þ(x)=√₁(x/a) 4/3 0 Find the electric field vector E and charge density p(x) between the plates. + p(x) a anode

A vacuum diode is shown at right. The heated cathode emits electrons into the region between two parallel plates, where they travel towards the anode. For a fixed anode voltage Vo, the system reaches a steady-state described by a static charge distribution p(x). The plates are separated by a distance a, and the electrostatic potential is cathode Þ(x)=√₁(x/a) 4/3 0 Find the electric field vector E and charge density p(x) between the plates. + p(x) a anode

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A vacuum diode is illustrated on the right. The heated cathode emits electrons into the region between two parallel plates, where they travel towards the anode. For a fixed anode voltage \( V_0 \), the system reaches a steady state described by a static charge distribution \( \rho(x) \). The plates are separated by a distance \( a \), and the electrostatic potential is given by \[ \Phi(x) = V_0 (x/a)^{4/3} \] The task is to find the electric field vector \( \mathbf{E} \) and charge density \( \rho(x) \) between the plates. **Diagram Explanation:** - The diagram shows two parallel plates, with the cathode on the left and the anode on the right. - Electrons are shown moving from the cathode to the anode, with a charge distribution \( \rho(x) \) between the plates. - A battery with voltage \( V_0 \) is connected across the plates, with the positive terminal connected to the anode. - The distance between the plates is denoted as \( a \). - The x-axis runs horizontally, indicating the direction of electron flow from the cathode to the anode.
Transcribed Image Text:A vacuum diode is illustrated on the right. The heated cathode emits electrons into the region between two parallel plates, where they travel towards the anode. For a fixed anode voltage \( V_0 \), the system reaches a steady state described by a static charge distribution \( \rho(x) \). The plates are separated by a distance \( a \), and the electrostatic potential is given by \[ \Phi(x) = V_0 (x/a)^{4/3} \] The task is to find the electric field vector \( \mathbf{E} \) and charge density \( \rho(x) \) between the plates. **Diagram Explanation:** - The diagram shows two parallel plates, with the cathode on the left and the anode on the right. - Electrons are shown moving from the cathode to the anode, with a charge distribution \( \rho(x) \) between the plates. - A battery with voltage \( V_0 \) is connected across the plates, with the positive terminal connected to the anode. - The distance between the plates is denoted as \( a \). - The x-axis runs horizontally, indicating the direction of electron flow from the cathode to the anode.
Expert Solution
Step 1

We have given a expression for potential.

We know that the electric field can be written as the negative gradient of potential and the other concepts is that the possion equation.

By using these two concepts we can solve the given problem as following in step two.

 

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