Q.What is the radius of the path of an electron (mass $9 \times 10^{-31}\ \text{kg}$ and charge $1.6 \times 10^{-19}\ \text{C}$) moving at a speed of $3 \times 10^{7}\ \text{m/s}$ in a magnetic field of $6 \times 10^{-4}\ \text{T}$ perpendicular to it? What is its frequency? Calculate its energy in keV. ($1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$).

Concept: charged particle in a $\perp$ magnetic field — circular motion.

Radius. The magnetic force supplies the centripetal force, $qvB=\dfrac{mv^2}{r}$, so

Frequency. $f=\dfrac{qB}{2\pi m}=\dfrac{(1.6\times10^{-19})(6\times10^{-4})}{2\pi(9\times10^{-31})}=\dfrac{9.6\times10^{-23}}{5.65\times10^{-30}}=1.70\times10^{7}\ \text{Hz}.$ …

Why it moves in a circle

The magnetic force $\vec F=q(\vec v\times\vec B)$ is always perpendicular to $\vec v$, so it does no work — the speed stays constant while the direction turns. For $\vec v\perp\vec B$ this force, of constant size $qvB$, acts as a centripetal force and the path is a circle.

1. Radius

Set the magnetic force equal to the centripetal force and cancel one $v$:

2. Frequency

The period is $T=\dfrac{2\pi r}{v}=\dfrac{2\pi m}{qB}$, so the frequency (independent of speed) is

3. Kinetic energy in keV …

Method: Lorentz Force & Circular Motion Analysis

This problem uses the Centripetal Force from Magnetic Lorentz Force method — when a charged particle enters a uniform magnetic field perpendicularly, the magnetic force provides the necessary centripetal force for circular motion.

Step 1: Find the radius of the circular path

The magnetic force on a moving charge is:

For circular motion, this equals the centripetal force:

Equating them:

Solving for radius $r$:

Substitute the values:

• $m = 9 \times 10^{-31}\ \text{kg}$
• $v = 3 \times 10^{7}\ \text{m/s}$
• $q = 1.6 \times 10^{-19}\ \text{C}$
• $B = 6 \times 10^{-4}\ \text{T}$

Step 2: Find the frequency of revolution

The time period for one complete revolution:

Frequency $f = \frac{1}{T}$:

Alternatively, using the direct formula (derived from $r$ expression):

Substitute:

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Here are the most common mistakes students make on this problem, why they happen, and how to avoid each one.

1. Forgetting the Perpendicular Condition

Mistake: Using the formula $r = \frac{mv}{qB}$ without checking if the velocity is perpendicular to the magnetic field.

Why it happens: Students often plug numbers into the formula without reading the phrase “perpendicular to it.”

How to avoid: Always underline the word perpendicular in the question. If the angle $\theta$ is not $90^\circ$, you must use $r = \frac{mv}{qB \sin \theta}$. Here, it’s given as perpendicular, so $\sin 90^\circ = 1$ — you’re safe.

2. Mixing Up Mass and Charge Values

Mistake: Using the mass of a proton ($1.67 \times 10^{-27}\ \text{kg}$) or the charge of an alpha particle ($3.2 \times 10^{-19}\ \text{C}$) instead of the electron’s values.

Why it happens: Many problems use similar numbers for different particles, and students rush.

How to avoid: Write down the given data clearly at the top:

• $m = 9 \times 10^{-31}\ \text{kg}$
• $q = 1.6 \times 10^{-19}\ \text{C}$
• $v = 3 \times 10^{7}\ \text{m/s}$
• $B = 6 \times 10^{-4}\ \text{T}$

Then double-check each value before substituting.

3. Incorrect Unit Conversion for Energy

Mistake: Computing energy in joules and then dividing by $1.6 \times 10^{-19}$ incorrectly, or forgetting that $1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$.

Why it happens: Students either invert the conversion factor or misplace the decimal.

How to avoid: Use the conversion as a multiplication:

Then convert eV to keV by dividing by $1000$:

4. Using the Wrong Formula for Frequency

Mistake: Using $f = \frac{v}{2\pi r}$ (which is for circular motion in general) but forgetting that in a magnetic field, the frequency is independent of speed.

Why it happens: Students derive frequency from radius and speed, which works but is inefficient and error-prone.

How to avoid: Use the cyclotron frequency formula directly:

This is faster and avoids carrying over errors from the radius calculation.

5. Arithmetic Errors with Powers of 10

Mistake: Adding or subtracting exponents incorrectly when multiplying or dividing numbers like $9 \times 10^{-31}$ and $1.6 \times 10^{-19}$.

Why it happens: Mental math under time pressure.

How to avoid: Write each step in scientific notation and separate the coefficients from the powers of 10: …