Question

Kirchhoff’s first law i.e. $ \sum i = 0 $ at a junction is based on the law of conservation of
(A) Charge
(B) Energy
(C) Momentum
(D) Angular momentum

Answer 1

Hint : Current is defined as the rate of passage of charge at a point with respect to time. Only the law of conservation of charge, amongst the options, gives a statement relating to charges. The law of conservation of charge states that the total sum of charges are equal before and after a process.

Formula used: In this solution we will be using the following formula;
 $ i = \dfrac{{dq}}{{dt}} $ where $ i $ is the instantaneous current flowing through a substance (or space), $ q $ is the charge which constitute the current, and $ t $ is time.
 $ \Delta q = 0 $ where $ \Delta q $ signifies a change in charge in a system.

Complete step by step answer
According to the principle of conservation of charge, the sum of the charges before and after a process are equal, that is to say the total charge is constant. Hence
 $ \Delta q = 0 $ where $ \Delta q $ signifies a change in charge in a system.
Now, when a charge is flowing through, say a conductor, the rate at which it flows, as known, is considered the current of flowing through the conductor. In average terms, it is given as
 $ I = \dfrac{{\Delta q}}{{\Delta t}} $
When we limit the time to zero, we get the instantaneous current is gotten as in,
 $ i = \mathop {\lim }\limits_{\Delta t \to 0} \dfrac{{\Delta q}}{{\Delta t}} = \dfrac{{dq}}{{dt}} $
Now, we need to imagine a conductor splitting into two conductors at a particular node, say A.
Since the charge does not undergo any change, $ \Delta q = 0 $ , then the instantaneous flow of current through the node also does not change.
Hence, $ \sum i = 0 $
Thus, the correct option is A.

Note
We should note that, although it is tempting to conclude that because $ \Delta q $ is zero, then $ \sum i = 0 $ , it is not so necessarily. For example, the current flowing into a box is not necessarily the current flowing out of the box since the charges could have been slowed down due to something (like say gases) in the box. Thus, even though the total charge coming out the box will still be the same as the charges going in, the current into the box is different from the current coming out. However, if we make this box so small that it is comparable to a dot, then the time used in this box is so small that no significant decrease could have happened, then the current is almost conserved. Now, if we limit the box to a dot (time to zero), mathematically, no decrease could have happened (principle of continuity) and the current is hence perfectly conserved. This is why we say the current into a node or at a junction (points with no dimension) is conserved.