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Question:

In a region, the potential is represented by V(x,y,z)=6x - xy - yz + 6yz, where V is in volts and x, y, z are in meters. The electric force experienced by a charge of 2 coulomb situated at point (1, 1, 1) is :

24N

6√5N

30N

4√35N

Solution:

V(x,y,z)=6x - xy - yz + 6yz
Ex=-∂V/∂x = -6 + y
Ey=-∂V/∂y = x + z
Ez=-∂V/∂z = y - 6y
→E=(-6+y)i + (x+z)j + (y-6y)k
→E(1,1,1)=-5i + 2j -5k
|→E|=√((-5)² + 2² + (-5)²) = √(25 + 4 + 25) = √54 = 3√6 N
C F=qE=2 x 3√6 = 6√6 N