# Electric flux and Gauss law

$tan(FAD) =DF/AF =4/3$, hence $FAD =53.13 degree$

angle between axis x and normal to the surface ABCD is

$alpha = 90-FAD =90-53.13 = 36.86 degree$

$tan(ADF) = 3/4$. hence $ADF =36.86 degree$

Hence angle between axis z and normal to the surface ABCD is

$beta =90- ADF = 53.13 degree$

Length of segment AD is $AD =sqrt{(3*3+4*4)} = 5$

Area of surface ABCD is $S =2*5 =10$

a) Electric flux through surface ABCD is

1. $Phi = S*Ex*cos(alpha) =10*ax*cos(36.86) = 8*ax$

2. $Phi =S*Ex*cos(alpha) =10*ay*cos(36.86) =8*ay$

3. $Phi=S*Ex*cos(alpha) =10*az*cos(36.86) =8*az$

4. $Phi = S*Ex*cos(alpha) = 10*ayz*cos(36.86) =8*ayz$

5. $Phi = S*Ex*cos(alpha) +S*Ez*cos(beta) =$

$=10*4ax*cos(36.86) +10*3ax*cos(53.13) =32*ax+18*ax =50*ax$

6. $Phi =S*Ex*cos(alpha) + S*Ez*cos(beta) =$

$= 10*(-3ax)*cos(36.86)+10*4ax*cos(53.13) =-24*ax+24*ax =0$

7.$Phi= S*Ex*cos(alpha) +S*Ey*cos(90) +S*Ez*cos(beta)=$ $=10*ax*cos(36.86)+0+10*az*cos(53.13) =8*ax+8*az$

Surface area $ABEF = S1 =2*3=6$

Surface area $CDEF = S2 =2*4 =8$

Surface area $BCE = AFD = S3 = 3*4/2 =6$

b) To tell why or why not there is a total charge inside the prism one needs to compute the total flux through the surfaces ABCD, ABEF, EFCD, BCE and AFD. As the Gauss law states this total flux is equal to the charge inside the prism divided by the space permittivity.

$Total flux = charge/epsilon_0$

1. Flux $CDEF =-S2*Ex =-8*ax$. Rest of above surfaces flux =0

Total flux =8ax-8ax =0. Hence there is no charge.

2. Flux $CDEF =-S2*Ex =-8*ay$. Rest of above surfaces flux =0

Total flux = -8ay+8*ay =0. hence there is no charge

3. Flux $CDEF = -S2*Ex =-8*az$. Total flux = -8az+8az =0. Hence there is no charge

4. Flux $CDEF = -S2*Ex = -8*ayz$. Total flux =-8ayz+8*ayz =0. Hence there is no charge

5. Flux $CDEF =-S2*Ex =-8*4ax =-32 ax$

Flux $ABEF = -S1*Ez =-6*3ax =-18*ax$

Total Flux =-32ax-18*ax+50 ax =0. Hence there is no charge

6. Flux $CDEF =-S2*Ex =8*(-3ax)=-24*ax$

Flux $ABEF =-S1*Ez =-6*4ax =-24*ax$

Total flux = -24ax-24ax+0 =0 There is charge inside the prism.

7 Flux $CDEF = -S2*Ex =-8*ax$

Flux $ABEF =-S1*Ez =-6*az$

Flux BEC=-Flux AFD

Total Flux =-8*ax-6*az +8ax+8az =2az There is charge inside the prism.

c. For the field lines please see attached file. In all three cases the lines are the same.