Problem 13.116 Challenge

a) จาก \(V(x, y, z) = -2xz - xy^{3} + 3yz^{2}\)

จะได้ \(V(1, -2, 0) = -2(1)(0) - (1)(-2)^{3} + 3(-2)(0)^{2} = 8.00 V \;\blacksquare\)

b) จาก \(\vec{E} = -\nabla V\)

îĵk̂

\(\vec{E} = -(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z})(-2xz - xy^{3} + 3yz^{2})\)

îĵk̂

\(\vec{E} = -((-2z - y^{3}) +(-3xy^{2} + 3z^{2}) +(-2x + 6yz) ) V/m\)

îĵk̂

\(\vec{E} = (2z + y^{3}) + (3xy^{2} - 3z^{2}) + (2x - 6yz) V/m \;\blacksquare\)

îĵ

\(c) \vec{E}(1, -2, 0) = (2(0) + (-2)^{3}) + (3(1)(-2)^{2} - 3(0)^{2})\)

k̂

\(+ (2(1) - 6(-2)(0)) V/m\)

îĵk̂

\(\vec{E}(1, -2, 0) = -8.00 + 12.0 + 2.00 V/m\)

\(\therefore |\vec{E}| =\sqrt{(-8.00)^{2} + (12.0)^{2} + (2.00^{2})}= 14.6 V/m \;\blacksquare\)

r̂îĵk̂

และ ทิศทาง \(= \frac{\vec{E}}{|\vec{E}|}= -0.548 + 0.822 + 0.137 \;\blacksquare\)