91影视

The electric field, $\stackrel{鈫�}{\mathrm{E}}=[鈭�3.00\widehat{\mathrm{i}}鈭�4.00{\mathrm{y}}^2\widehat{\mathrm{j}}+3.00\widehat{\mathrm{k}}]\text{鈥塏/颁}$

The edge length of the cube is$\mathrm{a}=2.00\text{鈥尘}$ with one corner at${\mathrm{x}}_1=5.00\text{鈥尘}$ ,${\mathrm{y}}_1=4.00\text{鈥尘}$

Using the gauss flux theorem, we can get the net flux through the surfaces. Now, using the same concept, we can get the net charge contained in the cube.

Formula:

The electric flux passing through the surface,

$\mathrm{蠒}=\mathrm{E}鈰�\mathrm{A}=\frac{\mathrm{q}}{{\mathrm{蔚}}_0}$ (1)

None of the constant terms will result in a nonzero contribution to the flux, so we focus on the x-dependent term only:

${\mathrm{E}}_{\mathrm{non}鈭�\mathrm{constant}}=鈭�4.00{\mathrm{y}}^2\widehat{\mathrm{i}}$

The face of the cube located at $\mathrm{y}=4.00\text{鈥�}\mathrm{m}$has an area $\mathrm{A}=4.00\text{鈥�}{\mathrm{m}}^2$(and it 鈥渇aces鈥� the +j direction) and has a 鈥渃ontribution鈥� to the flux that is given using equation (1) as:

$\begin{array}{c}{\mathrm{E}}_{\mathrm{non}鈭�\mathrm{constant}}\mathrm{A}=(3.00\text{鈥�}\mathrm{N}/\mathrm{C})(鈭�4.00脳{\left(4.00\text{鈥�}\mathrm{m}\right)}^2)\\ =鈭�256\text{鈥塏}鈰�{\text{m}}^{\text{2}}\text{/C}\end{array}$

The face of the cube located at $\mathrm{y}=2.00\text{鈥�}\mathrm{m}$has the same area A (however, this one 鈥渇aces鈥� the 鈥搄 direction) and a contribution to the flux that is given using equation (1) as:

$\begin{array}{c}鈭�{\mathrm{E}}_{\mathrm{non}鈭�\mathrm{constant}}\mathrm{A}=(鈭�4.00\text{鈥�}\mathrm{N}/\mathrm{C})(鈭�4.00脳{\left(2.00\text{鈥�}\mathrm{m}\right)}^2)\\ =64\text{鈥塏}鈰�{\text{m}}^{\text{2}}\text{/C}\end{array}$

Thus, the net flux is given using equations (a) and (b) as given:

$\begin{array}{c}\mathrm{蠒}=(鈭�256+64)\text{鈥塏}鈰�{\text{m}}^{\text{2}}\text{/C}\\ =鈭�192\text{鈥塏}鈰�{\text{m}}^{\text{2}}\text{/C}\end{array}$.

According to Gauss鈥檚 law, the net charge contained by the face is given as:

$\begin{array}{c}{\mathrm{q}}_{\mathrm{enc}}=(8.85脳{10}^{鈭�12}{\text{C}}^{\text{2}}/\text{N}鈰�{\text{m}}^{\text{2}})(鈭�192\text{鈥�}\text{N}鈰�{\text{m}}^{\text{2}}\text{/C})\\ =鈭�1.70脳{10}^{鈭�9}\text{C}\end{array}$

Hence, the value of the charge is $鈭�1.70脳{10}^{鈭�9}\text{C}$.