Answer
$$b = 2\frac{N}{Cm}$$
Work Step by Step
We are given a box-like Gaussian surface that encloses a net charge of +24.0$ε_{0}$ C with parameters $y_{2}$ = 1.00 m, $x_{1}$ = 1.00 m, $x_{2}$ = 4.00 m, $z_{1}$ = 1.00 m, $z_{2}$ = 3.00 m and lies in an electric field given by:
$$\vec{E}= [(10.0 + 2.00x)\hat{i} - 3.00\hat{j}+ bz\hat{k}] N/C$$
Where "b" is a constant and we need to find it.
The total flux through the box is given by Couloumb's Law:
$$\phi_{Total} = \int \vec{E} . d\vec{A} = \frac{q_{enc}}{\epsilon _{0}}$$
Hence we will need to find the flux through each side of the box and add all the fluxes to calculate the constant "b".
The flux through the x-axis will be the x component of the field multiplied by the area vector of the two surfaces in yz plane at $x_{1}$ and $x_{2}$:
$$\Phi_{x} = E_{x}(x_{2})A_{yz} - E_{x}(x_{1})A_{yz}$$
$$\Phi_{x} = A_{yz} [(1+2_{x2}) - (1+_{2x1})]$$
The area of the surface is: $$A_{yz} = y_{2}(z_{2} - z_{1})$$
Substituting:
$$\Phi_{x} = 2_{y2}(z_{2} - z_{1})(x_{2} - x_{1})$$
Substituting given values:
$$\Phi_{x} = 2(1)(2)(3) = 12 Nm^{2}/C$$
The flux through the y-axis will be the y component of the field multipled by the area vector of the two surfaces in xz plane at 0 and $y_{2}$:
$$\Phi_{y} = E_{y}A_{xz} - E_{y}A_{xz} = 0$$
The flux through the z-axis will be the z component of the field multiplied by the area vector of the two surfaces in xy plane at $z_{1}$ and $z_{2}$:
$$\Phi_{z}= E_{z}(z_{2})A_{xy} - E_{z}(z_{1})A_{xy}$$
$$\Phi_{z} = A_{xy}(b_{z2} - b_{z1})$$
$$\Phi_{z} = bA_{xy}(z_{2} - z_{1})$$
The area of the surface is:
$$A_{xy} = y_{2}(x_{2} - x_{1})$$
Substituting:
$$\Phi_{z} = b_{y2}(x_{2} - x_{1})(z_{2} - z_{1})$$
Substituting the values:
$$\Phi_{z} = b(1)(3)(2) = 6b$$
Therefore the total flux will be:
$$\Phi_{total} = \Phi_{x} + \Phi_{y} + \Phi_{z}$$
$$\Phi_{Total} = 12 + 0 + 6b = 12 + 6b$$
Substituting in Couloumb's law:
$$\Phi_{Total} = \frac{q_{enc}}{\epsilon _{0}}$$
$$12 + 6b = \frac{q_{enc}}{\epsilon_{0}}$$
Putting the value of $q_{enc}$:
$$12 + 6b = \frac{24\epsilon_{0}}{\epsilon_{0}}$$
$$12 + 6b = 24$$
Therefore,
$$b = 2$$
Finding the unit:
Since "bz" is an electric field, its units are N/C
Therefore:
$$bz = \frac{N}{C}$$
We are given that "z" is in meters:
$$b(m) = \frac{N}{C}$$
Therefore:
$$b = 2\frac{N}{Cm}$$
