Question

Maxwell’s Four Equations are written as
(i) $\int \overset{\to }{\mathop{E}}\,.\overset{\to }{\mathop{ds}}\,=\dfrac{{{q}_{0}}}{{{E}_{0}}}$
(ii) $\int \overset{\to }{\mathop{B}}\,.\overset{\to }{\mathop{ds}}\,=0$
(iii) \[\int \overset{\to }{\mathop{E}}\,.\overset{\to }{\mathop{dl}}\,=\dfrac{d}{dt}\int \overset{\to }{\mathop{B}}\,.\overset{\to }{\mathop{ds}}\,\]
(iv) \[\int \overset{\to }{\mathop{B}}\,.\overset{\to }{\mathop{ds}}\,={{\mu }_{0}}{{\varepsilon }_{0}}\dfrac{d}{dt}\int \overset{\to }{\mathop{E}}\,.\overset{\to }{\mathop{ds}}\,\]
Which of the above Maxwell's equation 1 shows that the electric field lines do not form closed loops.
A) (i) only
B) (iii) only
C) (ii) only
D) (iv) only

Answer 1

Hint: Look for the equation which can lead you to the divergence of the vector. For closed lines, the divergence of the vector must be equal to zero. In the given problem we are examining electric field lines, so focus on the equation depicting divergence of electric field vectors. As electric field lines don't form closed loops, the divergence of their vector must not be zero.

Complete step by step solution:
We need to find out which Maxwell's equation 1 shows that the electric field lines do not form closed loops.

The Maxwell equations help us to examine the electric and magnetic fields due to electric charges and currents.

Equation 1, i.e. $\int \overset{\to }{\mathop{E}}\,.\overset{\to }{\mathop{ds}}\,=\dfrac{{{q}_{0}}}{{{E}_{0}}}$
indicates that the closed surface integral of electric flux density is equal to the charge enclosed by that closed surface.

Applying gauss divergence theorem in equation 1 will lead to
\[\nabla .E=\dfrac{\sigma }{{{\varepsilon }_{0}}}\]
But for closed field lines, \[\nabla .E=0\] which is not the fact.

For example, if we take a positive electric charge, the electric field lines are radially outwards. The divergence of electric field vectors, \[\nabla .E\], is not equal to zero which indicates the existence of electric monopoles. As the electric field lines do not originate and terminate in the same charge body, their divergence is not zero.

Conceptually, the electric field lines do not reside inside the charge, hence, they never form closed loops as depicted by Maxwell equation 1.

Note:
As we are concerned only with electric field lines, option b can be eliminated directly. Equation 3 and 4, clearly shows both electric and magnetic field vectors, hence, indicating their relationship with each other. Therefore, these options can also be eliminated, which gives us the solution i.e., option 1.