In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius 0.5 A. If t

Q: Why does a single electron create a current?

Even a single moving charge constitutes an electric current because current is simply the rate of charge passing a given point. In a circular orbit, the electron passes the same point repeatedly, creating a steady average current.

Q: Why do we use the time period $t$ in the denominator?

The time period $t$ is the time it takes for the electron to complete one full loop. If you stand at one point on the orbit, a charge $e$ passes you every $t$ seconds. Thus, the average rate of charge flow is $e/t$.

Q: What is an Angstrom ($\text{\AA}$)?

An Angstrom is a unit of length equal to $10^{-10}$ meters. It is commonly used to express atomic dimensions.

Q: Why did we substitute $\pi$ as $22/7$?

The problem explicitly instructs us to use $\pi = 22/7$. This often helps in simplifying the calculation, as the $2.2$ in the numerator cancels nicely with the $22$ in the denominator.

Q: How do we convert Amperes to milliamperes (mA)?

One Ampere is equal to $1000$ milliamperes ($1 \text{ A} = 10^3 \text{ mA}$). So, you multiply the value in Amperes by $10^3$ to get the value in mA.

Question

In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius $0.5 	ext{ \AA}$. If the speed of electron is $2.2 	imes 10^6 	ext{ m/s}$, then the current associated with the electron will be ............... $	imes 10^{-2} 	ext{ mA}$. [Take, $\pi$ as $\frac{22}{7}$]

Sigma Percentile · Accepted Answer

### The Microscopic Current Loop

Imagine an electron zooming around the nucleus in a circular orbit, just like a planet orbiting the sun. While we usually think of current as a stream of countless electrons flowing through a wire, even a single moving charge constitutes an electric current! This is because current is fundamentally defined as the rate of flow of charge past a given point.

### Formulating the Equivalent Current

To find this equivalent current, we start with the basic definition of current:

$I = \frac{q}{t}$

For an electron completing one full revolution, the total charge $q$ that passes a point on the orbit is simply the elementary charge $e$. The time $t$ it takes to complete this revolution is the time period.

We know from basic kinematics that time is distance divided by speed. The distance traveled in one revolution is the circumference of the orbit, $2\pi r$, and the speed is $v$. Therefore, the time period is:

$t = \frac{2\pi r}{v}$

Substituting this time period back into our current equation, the velocity $v$ flips up to the numerator, giving us a beautiful and compact formula for the equivalent current of a revolving charge:

$I = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r}$

### Crunching the Numbers

Now, we just need to carefully substitute the values given in the problem. We are given the radius $r = 0.5 	ext{ \AA}$, which must be converted to standard SI units: $0.5 	imes 10^{-10} 	ext{ m}$. The speed is $v = 2.2 	imes 10^6 	ext{ m/s}$, and the charge of an electron is $e = 1.6 	imes 10^{-19} 	ext{ C}$. The problem also instructs us to use $\pi = \frac{22}{7}$.

Let's plug these into our master equation:

$I = \frac{1.6 	imes 10^{-19} 	imes 2.2 	imes 10^6}{2 	imes \frac{22}{7} 	imes 0.5 	imes 10^{-10}}$

**Don't rush the calculation!** Let's simplify the denominator first. Notice how $2 	imes 0.5 = 1$. This leaves us with just $\frac{22}{7} 	imes 10^{-10}$ in the denominator.

Now, let's gather the numbers and the powers of $10$ separately:

$I = \frac{1.6 	imes 2.2 	imes 7}{22} 	imes 10^{-19 + 6 + 10}$

$I = \frac{24.64}{22} 	imes 10^{-3} = 1.12 	imes 10^{-3} 	ext{ A}$

### The Final Conversion

We have our current in Amperes, but the question sets a specific trap. It asks for the answer in the format of $10^{-2} 	ext{ mA}$.

First, let's convert Amperes to milliamperes (mA) by multiplying by $10^3$:

$I = 1.12 	imes 10^{-3} 	imes 10^3 	ext{ mA} = 1.12 	ext{ mA}$

Finally, to match the requested format of $	imes 10^{-2} 	ext{ mA}$, we shift the decimal point two places to the right:

$I = 112 	imes 10^{-2} 	ext{ mA}$

Thus, the integer value we are looking for is **112**.

Enter Numerical Value:

Visualized Solution

Visualizing the Orbit

Definition of Current

Time Period of Revolution

Equivalent Current Formula

Substituting the Values

Calculating the Current

Final Unit Conversion

The Way Forward

The Sigma Insight: Bohr's Atomic Model and Energy Levels

The Microscopic Current Loop

Formulating the Equivalent Current

Crunching the Numbers

The Final Conversion

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