Concept:The magnetic force on a charge moving at an angle \( \theta \) to a magnetic field is given by the Lorentz force equation.
Formula:$$ F = qvB \sin(\theta) \implies q = \frac{F}{vB \sin(\theta)} $$
Solution:- List the variables: \( F = 2.78 \times 10^{-17} \ \text{N} \), \( v = 10 \ \text{m/s} \), \( B = 10 \ \text{T} \), \( \theta = 60^\circ \).
- Recall that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \).
- Calculate the denominator: \( vB \sin(60^\circ) = (10)(10)(0.866) = 100 \times 0.866 = 86.6 \).
- Divide the force by this value: \( q = \frac{2.78 \times 10^{-17}}{86.6} = 0.0321 \times 10^{-17} \ \text{C} \).
- Convert to scientific notation: \( q \approx 3.20 \times 10^{-19} \ \text{C} \) (which is exactly the charge of an alpha particle or two protons).
Why other options are incorrect:Option A is the charge of a single electron/proton. Options B and C are mathematically incorrect derivations that occur if you fail to use the sine function or multiply incorrectly.
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