Hand write ，more details， step by step solutions 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.

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