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Step-by-Step Solution
Step 1: List the known quantities
• Charge of the particle,
q = 2 \times 10^{-6}\,\mathrm{C}
• Potential difference,
V = 100\,\mathrm{V}
• Magnetic field strength,
B = 4 \times 10^{-3}\,\mathrm{T}
• Radius of the semicircle,
r = 3 \times 10^{-2}\,\mathrm{m}
Step 2: Relate kinetic energy and potential difference
The charged particle gains kinetic energy by being accelerated through a potential difference V . Hence,
\frac{1}{2}\,m\,v^2 = q\,V.
From this, the velocity v can be written as
v = \sqrt{\frac{2\,q\,V}{m}}.
Step 3: Apply the magnetic force condition
When a charge moves perpendicular to a uniform magnetic field, the magnetic force acts as the centripetal force:
q\,v\,B = \frac{m\,v^2}{r}.
Rearranging,
r = \frac{m\,v}{q\,B}.
Substitute v = \sqrt{\frac{2\,q\,V}{m}} into the above expression:
r = \frac{m\,\sqrt{\frac{2\,q\,V}{m}}}{q\,B}
= \frac{\sqrt{2\,m\,q\,V}}{q\,B}.
Step 4: Solve for mass m
Square both sides:
r^2 = \left(\frac{\sqrt{2\,m\,q\,V}}{q\,B}\right)^2
= \frac{2\,m\,q\,V}{(q\,B)^2}.
So,
r^2\,(q\,B)^2 = 2\,m\,q\,V
\quad\Longrightarrow\quad
m = \frac{r^2\,(q\,B)^2}{2\,q\,V}.
Since one factor of q cancels, this becomes
m = \frac{r^2\,q\,B^2}{2\,V}.
Step 5: Substitute the numerical values
• r = 3 \times 10^{-2}\,\mathrm{m}
• q = 2 \times 10^{-6}\,\mathrm{C}
• B^2 = \left(4 \times 10^{-3}\right)^2 = 16 \times 10^{-6}\,\mathrm{T}^2
• V = 100\,\mathrm{V}
Hence,
m = \frac{(3 \times 10^{-2})^2 \times (2 \times 10^{-6}) \times (4 \times 10^{-3})^2}{2 \times 100}.
Step 6: Perform the arithmetic
1) Combine the numeric coefficients:
9 \times 10^{-4} \times 2 \times 16 = 288 \times 10^{-4 - 6 - 6} = 288 \times 10^{-16}.
2) Thus the numerator is 288 \times 10^{-16} = 2.88 \times 10^{-14}.
3) Divide by 200 = 2 \times 10^2:
m = \frac{2.88 \times 10^{-14}}{2 \times 10^2} = 1.44 \times 10^{-16}\,\mathrm{kg}.
Step 7: Express in the desired form
To express m in the form " \ldots \times 10^{-18} \,\mathrm{kg} ", multiply by 100:
m = 144 \times 10^{-18}\,\mathrm{kg}.
Final Answer
The mass of the charged particle is
144 \times 10^{-18}\,\mathrm{kg}.
