To solve the given problem of calculating the work done by a force as a particle moves from \( x = 0 \) to \( x = x_1 \), we need to understand the concept of work done by a variable force.
The force acting on the particle is given by \( F = Cx \), where \( C \) is a constant and \( x \) is the position of the particle.
The work done by a variable force when moving an object from position \( x_0 \) to position \( x_1 \) is given by the integral:
\(W = \int_{x_0}^{x_1} F(x) \, dx\)
In this case, we have:
\(W = \int_{0}^{x_1} Cx \, dx\)
Carrying out this integration:
\(W = C \int_{0}^{x_1} x \, dx = C \left[ \frac{x^2}{2} \right]_{0}^{x_1}\)
Evaluating the integral, we get:
\(W = C \left( \frac{x_1^2}{2} - \frac{0^2}{2} \right)\)
\(W = C \cdot \frac{x_1^2}{2}\)
Therefore, the work done by the force as the particle moves from \( x = 0 \) to \( x = x_1 \) is:
\(\frac{1}{2} C x_1^2\)
This matches the given correct answer option: \( \frac{1}{2} C x_1^2 \).
Conclusion: The correct answer is \(\frac{1}{2} C x_1^2\). This is based on applying the concept of work done by a variable force and accurately evaluating the integral for this specific force function.