Question

In Problems 75 through 77, you are given the equation(s) used to solve a problem. For each of these, you are to: a. Write a realistic problem for which this is the correct equation (s). b. Finish the solution to the problem.

$2az\mathrm{V}/\mathrm{m}=-\frac{dV}{dz}$whereas a is a constant with units of$V/m^{2}V(z=0)=10V$.

Short answer

(a) The electric field is given as 2az, where a is a constant and z is the distance, in meters. Find the potential if the potential at distance z = 0 is 10 V. "

(b). $V\left(z\right)=10-a{z}^2$

Step-by-step solution

Step 1. Given information

$2az\mathrm{V}/\mathrm{m}=-\frac{dV}{dz}$, where a is a constant with units of$V/m^2,V(z=0)=10V.$

Step 1. Finding a realistic problem for part (a)

Part (a)

The electric field is given as 2az, where a is a constant and z is the distance, in meters. Find the potential if the potential at distance z=0 is 10 V. "

Step 3. Simplify for Part (b).

Part (b)

We know that the following holds true:

$2az=-\frac{dV}{dz}$

We can separate and integrate on both sides, thus finding

$\int dV=-\int 2azdz$

Which leaves us

$V\left(z\right)=-a{z}^2+c$

The integration constant can be found by knowing the initial condition.

$V\left(0\right)=c=10$

This means that the potential is given as a function of z as

localid="1649476813420" $V\left(z\right)=10-a{z}^2$.