The electric field components in are Ex = A+Bx, Ey = Ez = 0, in which A = 800 N/C , B= 100 N/C-m. Calculate 
(a) the flux through the cube
(b) the charge within the cube. Assume that a = 0.1 m

Asked by hitanshu04 | 22nd May, 2021, 07:41: PM

Expert Answer:


Let us assume that the cube is positioned in Cartesian coordinate system so that centre of the cube
is at the origin and sides are parallel to the axes.
 
Hence the face ABCD is at x = +a/2  and the face EFGH is at x = -a/2as shown in figure.
 
Flux φ passing through the surface of cube is determined as
 
begin mathsize 14px style phi space equals space surface integral space E with rightwards arrow on top times space stack d A with rightwards arrow on top end style
where begin mathsize 14px style E with rightwards arrow on top end style is electric field vector , dA is area element and integration is performed over the closed surface of cube
Since electric field is only in x-axis direction , Flux is determined as
 
begin mathsize 14px style phi space equals space double integral E subscript 1 x end subscript space d y d z space minus space double integral E subscript 2 x end subscript d y d z space end style............................... (1)
where E1x is electric field at face ABCD , i.e., at x = +a/2 , E2x is electric field at face EFGH , i.e., at x = -a/2
 
First term in eqn.(1) is positive because unit normal vecrtor is in +x axis direction .
 
Second term in eqn.(1) is negative because unit normal vecrtor is in -x axis direction .
 
E1x = ( A + 0.05 × B )  
 
E2x = ( A - 0.05 × B )
 
Hence eqn.(1) is written as ,
 
begin mathsize 14px style phi space equals space double integral left parenthesis A space plus space 0.05 space cross times space B space right parenthesis space d y d z space minus space double integral left parenthesis A space minus space 0.05 space cross times space B space right parenthesis d y d z space end style
 
begin mathsize 14px style phi space equals space space left parenthesis 0.1 cross times B space right parenthesis double integral d y d z space equals 100 space cross times space 0.1 space cross times 0.1 space cross times space 0.1 space equals space 0.1 space N space C to the power of negative 1 end exponent m squared end style
( surface integration will give the face area of the cube )
 
By Gauss law , enclosed charge Q is related to the surface flux φ as ,   φ = Q / εo
 
where εo is permittivity of free space
 
Enclosed charge Q = ( φ × ε ) = ( 0.1 × 8.854 × 10-12 ) C = 8.854 × 10-11 C

Answered by Thiyagarajan K | 23rd May, 2021, 12:47: AM