2 The potential function V in a certain electrostatic system is given by V 20003 r2 2 z2 volts in cylindrical coordinats Find the equipotential surface passing by the point ft r 2 m 30 z 1 m Find also...

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(2) The potential function $V$ in a certain electrostatic system is given by \[ V=\frac{2000}{3+r^{2} \sin ^{2} \varphi+z^{2}} \text { volts(in cylindrical coordinats) } \] Find the equipotential surface passing by the point $\left(\mathrm{r}=2 \mathrm{~m}, \Phi=30^{\circ}, \mathrm{z}=1 \mathrm{~m}\right)$ Find also the potential gradient and the electric field at this point.

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#### Solution By Steps ***Step 1: Identify the Equipotential Surface*** To find the equipotential surface passing through the given point, set $V$ equal to the potential at that point and solve for $r$, $\varphi$, and $z$. ***Step 2: Substitute the Given Values*** Substitute $r=2$ m, $\varphi=30^\circ$, and $z=1$ m into the potential function. ***Step 3: Solve for the Equipotential Surface*** Calculate the value of the potential at the given point to determine the equation of the equipotential surface. ***Step 4: Find the Potential Gradient*** Calculate the gradient of the potential function $V$ with respect to $r$, $\varphi$, and $z$. ***Step 5: Calculate the Electric Field*** The electric field $\vec{E}$ is the negative gradient of the potential function. Calculate $\vec{E}$ at the given point using the gradient of $V$. #### Final Answer The equation of the equipotential surface passing through the point is $r^2\sin^2\varphi + z^2 = 2000$. The potential gradient is $\nabla V = \left(\frac{-4000r\sin^2\varphi}{(3+r^2\sin^2\varphi+z^2)^2}, \frac{0}{(3+r^2\sin^2\varphi+z^2)^2}, \frac{-4000z}{(3+r^2\sin^2\varphi+z^2)^2}\right)$. The electric field at the point $\left(r=2 \text{ m}, \varphi=30^\circ, z=1 \text{ m}\right)$ is $\vec{E} = \left(\frac{-1600\sqrt{3}}{2001^2}, 0, \frac{-400}{2001^2}\right)$ V/m. #### Key Concept Equipotential Surface #### Key Concept Explanation Equipotential surfaces are surfaces in a field where the potential function has a constant value. They help visualize regions of equal potential in electric fields, aiding in understanding the field's behavior and the direction of the electric field lines.

Follow-up Knowledge or Question

What is an equipotential surface in an electrostatic system and how is it related to the potential function?

How can we mathematically express the potential gradient in an electrostatic system?

How is the electric field related to the potential gradient in an electrostatic system?

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