What is the electric field at (x,y,z) = (1,1,1) if the electric potential is known to be V(x,y,z) = ax4y + by3(z-c) + dx2z, where a, b, c and d are positive constants?
1. What is the electric field at (x,y,z) = (1,1,1) if the electric potential is known to be V(x,y,z) = ax4y + by3(z-c) + dx2z, where a, b, c and d are positive constants?
2. We choose the origin to have a potential of zero, V(0,0,0) = 0. What other points have zero potential?
Potential energy: U(r)=kqQr,ΔU=qΔV
Potential difference: ΔVAB=VB−VA=−∫ABE→⋅dl→=−Wq(A→B)q
Electric potential: V=kqr=k∫dqr
The electric field from potential: E→=−∇→V=−i^∂V∂x−j^∂V∂y−k^∂V∂z=−r^∂V∂r−θ^1r∂V∂θ−ϕ^1rsinθ∂V∂ϕfor planar and spherical coordinates, respectively.
Surface Area of a cylinder: A=2πrh+2πr2
Surface Area of a sphere: A=4πr2
Volume charge density: ρ=chargeQvolumeV
Surface charge density: σ=chargeQareaA
Linear charge density: λ=chargeQlengthL
Differential volume: dV=dxdydz=r2sinθdrdθdϕ=rdzdrdθ for planar, spherical, and cylindrical coordinates, respectively.
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