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Step-by-Step Solution
Step 1: Identify the forces acting on the pendulum bob
The pendulum bob (mass $2\,\text{g} = 2 \times 10^{-3}\,\text{kg}$, charge $5.0\,\mu\text{C} = 5 \times 10^{-6}\,\text{C}$) is influenced by:
Its weight, $mg$, acting vertically downward.
The tension $T$ in the string, acting along the string.
The electric force $qE$, acting horizontally (since the electric field is horizontal).
Step 2: Resolve tension into components
At equilibrium, the bob remains at a constant angle $\theta$ with the vertical, so:
The vertical component of tension balances the weight:
$$ T \cos \theta = mg. $$
The horizontal component of tension balances the electric force:
$$ T \sin \theta = qE. $$
Step 3: Express $\tan \theta$
Divide the second equation by the first:
$$
\tan \theta = \frac{T \sin \theta}{T \cos \theta}
= \frac{qE}{mg}.
$$
Step 4: Substitute the given values
Plug in $q = 5 \times 10^{-6}\,\text{C}$, $E = 2000\,\text{V/m}$, $m = 2 \times 10^{-3}\,\text{kg}$, and $g = 10\,\text{m/s}^2$:
$$
\tan \theta = \frac{(5 \times 10^{-6}) \times (2000)}{(2 \times 10^{-3}) \times 10}.
$$
Evaluate the numerator and denominator separately:
Numerator: $5 \times 10^{-6} \times 2000 = 0.01\,\text{N}.$
Denominator: $(2 \times 10^{-3}) \times 10 = 0.02\,\text{N}.$
Hence,
$$
\tan \theta = \frac{0.01}{0.02} = 0.5.
$$
Step 5: Find the angle $\theta$
Taking the inverse tangent,
$$
\theta = \tan^{-1}(0.5).
$$
This matches the correct answer:
$\theta = \tan^{-1}(0.5).$
