# Given that D=2x(1+z^2)a_x + 2x^2za_z nC/m2. a) Find p_v b) Determine the flux crossing the...

## Question:

Given that {eq}D=2x(1+z^2)a_x + 2x^2za_z {/eq} {eq}nC/m2 {/eq}.

a) Find {eq}p_v {/eq}.

b) Determine the flux crossing the rectangular region defined by {eq}z=1 {/eq}, {eq}0 < x < 2 {/eq}, {eq}0 < y < 3 {/eq}.

## Magnetic flux:

It defines as the quantity of magnetic field traveling through the surface of a conducting coil. Mathematically it is the product of the magnetic field and the area which is orthogonal to the magnetic field.

Part (a):

The expression for the {eq}{p_v} {/eq} is,

{eq}{p_v} = \Delta \cdot D {/eq}..... (1)

Here, the equation of {eq}D {/eq} is given by,

{eq}D = 2x\left( {1 + {z^2}} \right){a_x} + 2{x^2}z{a_z} {/eq}..... (2)

Substitute the value of equation (2) in equation (1)

{eq}\begin{align*} {p_v} &= \Delta \cdot \left\{ {2x\left( {1 + {z^2}} \right){a_x} + 2{x^2}z{a_z}} \right\}\\ {p_v} &= \dfrac{\partial }{{\partial x}}\left[ {2x\left( {1 + {z^2}} \right)} \right] + \dfrac{\partial }{{\partial z}}\left( {2{x^2}z} \right)\\ {p_v} &= \left[ {2\left( {1 + {z^2}} \right) + 2{x^2}} \right]\;{\rm{nC/}}{{\rm{m}}^{\rm{3}}} \end{align*} {/eq}

Thus, the value of {eq}{p_v} {/eq} is {eq}\left[ {2\left( {1 + {z^2}} \right) + 2{x^2}} \right]\;{\rm{nC/}}{{\rm{m}}^{\rm{3}}} {/eq}

Part (b):

The expression for the flux which is given by coulomb's law,

{eq}\varphi = \oint\limits_s {D \cdot ds} {/eq} .......... (3)

Here, {eq}ds = dxdydz {/eq}

Substitute the values in equation (3)

{eq}\begin{align*} \varphi &= \int\limits_{y = 0}^3 {\int\limits_{x = 0}^2 {2{x^2}zdxdy} } \\ \varphi &= \int\limits_0^3 {\left( {\int\limits_0^2 {{x^2}dx} } \right)dy} \end{align*} {/eq}

Solve the limit,

{eq}\begin{align*} \varphi &= 2 \times \dfrac{1}{3}\left( {{x^3}} \right)_0^2 \times \left( y \right)_0^3\\ \varphi &= 2 \times \dfrac{8}{3} \times 3\\ \varphi &= 16\;{\rm{nC}} \end{align*} {/eq}

Thus, the flux crossing the rectangular region is, {eq}16\;{\rm{nC}} {/eq}.