Problem 13.117 Challenge
หัวข้อ: การหาสนามไฟฟ้าจากศักย์ไฟฟ้า (Obtaining the Value of the Electric Field from the Electric Potential)
\(\therefore |\vec{E}| =\sqrt{(7.00)^{2} + (-2.00)^{2} + (8.00^{2})}= 10.8 V/m \;\blacksquare\)
ดูวิธีทำ
a) จาก \(V(x, y, z) = 5x - 3xy^{2} + 2yz^{2}\)
จะได้ \(V(0, 2, -1) = 5(0) - 3(0)(2)^{2} + 2(2)(-1)^{2} = 4.00 V \;\blacksquare\)
b) จาก \(\vec{E} = -\nabla V\)
îĵk̂
\(\vec{E} = -(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z})(5x - 3xy^{2} + 2yz^{2})\)
îĵk̂
\(\vec{E} = -((5 - 3y^{2}) +(-6xy + 2z^{2}) +(4yz) ) V/m\)
îĵk̂
\(\vec{E} = (3y^{2} - 5) + (6xy - 2z^{2}) - 4yz V/m \;\blacksquare\)
îĵk̂
\(c) \vec{E}(0, 2, -1) = (3(2)^{2} - 5) + (6(0)(2) - 2(-1)^{2}) - 4(2)(-1) V/m\)
îĵk̂
\(\vec{E}(1, -2, 0) = 7.00 - 2.00 + 8.00 V/m\)
\(\therefore |\vec{E}| =\sqrt{(7.00)^{2} + (-2.00)^{2} + (8.00^{2})}= 10.8 V/m \;\blacksquare\)