Question
In a region, the potential is represented by $$V\left( {x,y,z} \right) = 6x - 8xy - 8y + 6yz,$$ where $$V$$ is in volts and $$x,y,z$$ are in metres. The electric force experienced by a charge of $$2C$$ situated at point $$\left( {1,1,1} \right)$$ is
A.
$$6\sqrt 5 N$$
B.
$$30\,N$$
C.
$$24\,N$$
D.
$$4\sqrt {35} \,N$$
Answer :
$$4\sqrt {35} \,N$$
Solution :
As we know that relation between potential difference and electric field $$E$$ in a particular region is given by, $$E = - \frac{{dV}}{{dr}}$$
As, $$V = 6x - 8xy - 8y + 6yz$$
So, $$E = - \frac{{dV}}{{dr}}$$
$$ = - \left[ {\left( {6 - 8y} \right)\hat i + \left( { - 8x - 8 + 6z} \right)\hat j + 6y\hat k} \right]$$
The value of $$E$$ at coordinate $$\left( {1,1,1} \right)$$
$$\eqalign{
& E = - \left[ { - 2\hat i - 10\hat j + 6\hat k} \right] \cr
& {\text{So,}}\,\,{E_{{\text{net}}}} = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( { - 10} \right)}^2} + {6^2}} = 2\sqrt {35} \,N/C \cr} $$
and force on charge $$q$$ due to $${E_{{\text{net}}}}$$ is given by
$$F = q{E_{{\text{net}}}} = 2 \times 2\sqrt {35} = 4\sqrt {35} \,N$$