Can someone please help me with this? Let M be the closed surface that consists

Question

Can someone please help me with this?

Let M be the closed surface that consists of the hemisphere M_1: x^2 + y^2 + z^2 = 1, z greaterthanorequalto 0, M_2: x^2 + y^2 lessthanorequalto 1, z = 0 and its base Let E be the electric field defined by E = (20x, 20y, 20z). Find the electric flux across M. Write the integral over the hemisphere using spherical coordinates, and use the outward pointing normal. Doubleintegral_M_1 E middot dS = integral_a^b integral_c^d f(theta, phi) d theta d phi, Where Using t for theta and p for phi.

Explanation / Answer

M_1: Use r(, ) = <cos sin , sin sin , cos > for in [0, 2], in [0, /2].

Since r_ x r_ = <-sin^2() cos , -sin^2() sin , -sin cos >,
n = <sin^2() cos , sin^2() sin , sin cos >, for the normal to point out of M_1.

So, (M_1) <8x, 8y, 8z> · dS
= ( = 0 to /2) ( = 0 to 2) <8 cos sin , 8 sin sin , 8 cos > ·
<sin^2() cos , sin^2() sin , sin cos > d d

= ( = 0 to /2) ( = 0 to 2) [8 sin^3() + 8 cos^2() sin ] d d
= ( = 0 to /2) ( = 0 to 2) [8 sin^2() + 8 cos^2()] sin d d
= ( = 0 to /2) ( = 0 to 2) 8 sin d d
= -16 cos {for = 0 to /2}
= 32.
------------------
M_2: Use r(u, v) = <u cos v, u sin v, 0> for u in [0, 1], v in [0, 2].

Since r_u x r_v = <0, 0, u>
(M_2) <8x, 8y, 8z> · dS
= (u = 0 to 1) (v = 0 to 2) <8u cos v, 8u sin v, 0> · <0, 0, u> dv du
= 0.
--------------------
Therefore, M E · dS = 32 + 0 = 32.

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---