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Question: An electron is moving in a circle of radius \(5.1\times {{10}^{-11}}m\) at a frequency of \(6.8\time...

An electron is moving in a circle of radius 5.1×10−11m5.1\times {{10}^{-11}}m at a frequency of 6.8×1015revolution/s6.8\times {{10}^{15}}revolution/s. The equivalent current is approximately :-
A. 5.1×10−3A5.1\times {{10}^{-3}}A
B. 6.8×10−3A6.8\times {{10}^{-3}}A
C. 1.1×10−3A1.1\times {{10}^{-3}}A
D. 2.2×10−3A2.2\times {{10}^{-3}}A

Explanation

Solution

Current is equal to the total charge flown through a point (or cross section) in a given time divided by the given time. So, calculate the time period of the motion of the electron and divide charge by the time period.

Formula used:
i=qti=\dfrac{q}{t}
T=1fT=\dfrac{1}{f}

Complete step by step answer:
An electric current is defined as the amount of charge flowing through a cross section per unit time.

In this case, an electron is moving along a circular path and we know that an electron possesses a large amount of charge. Therefore, there is a movement of charge along time. In other words, there is some charge flowing along the circular path.

As said above current is equal to the amount of charge flowing per unit time. This means that a constant current (i) is equal to the total charge flown (q) through a point (or cross section) in a given time (t) divided by the given time.

i.e. i=qti=\dfrac{q}{t}.

Since the electron is moving in a circular path, it will have a time period (T). In time equal to the time period, the electron will complete in revolution.

Therefore, from a point the charge flown in time T is equal to the charge on the electron. (i.e.
1.6×10−19C1.6\times {{10}^{-19}}C).

This means that the current in the case is i=1.6×10−19CTi=\dfrac{1.6\times {{10}^{-19}}C}{T} …. (i).

The time period is given as T=1fT=\dfrac{1}{f} …. (ii), where f is the frequency of the motion. It is given that f=6.8×1015revolution/s=6.8×1015×2πs−1f=6.8\times {{10}^{15}}revolution/s=6.8\times {{10}^{15}}\times 2\pi {{s}^{-1}}.

Substitute the value of f in (ii).
⇒T=16.8×1015×2πs\Rightarrow T=\dfrac{1}{6.8\times {{10}^{15}}\times 2\pi }s.
Substitute the value of T in (i).
⇒i=1.6×10−19C16.8×1015×2π=1.6×10−19×6.8×1015×2π=68.36×10−4A≈6.8×10−3A\Rightarrow i=\dfrac{1.6\times {{10}^{-19}}C}{\dfrac{1}{6.8\times {{10}^{15}}\times 2\pi }}=1.6\times {{10}^{-19}}\times 6.8\times {{10}^{15}}\times 2\pi =68.36\times {{10}^{-4}}A\approx 6.8\times {{10}^{-3}}A.

Hence, the correct option is B.

Note: In the above case, we can say that the value of current is constant because there is only one electron and it is moving with a uniform speed.

Although we assumed the current in this case to be a constant value, it is not the charge flowing through a point is not continuous. At a given point the charge flows after every time equal to T.
What we calculated is the average current in time equal to T.

As a result, the instantaneous current at a point may be zero or non-zero at a given time.