A thin rod of length L and total charge Q has the nonuniform linear charge distribution λ(x)=λ ₀x/L, where x is measured from the rod's left end. a. What is λ₀ in terms of Q and L?

Verified step by step guidance

1. Understand the given charge distribution function, \( \lambda(x) = \frac{\lambda_0 x}{L} \), where \( \lambda_0 \) is a constant to be determined.
2. Recognize that the total charge Q on the rod can be found by integrating the charge density \( \lambda(x) \) over the length of the rod from 0 to L.
3. Set up the integral for the total charge: \( Q = \int_0^L \lambda(x) \, dx \). Substitute the expression for \( \lambda(x) \) into the integral: \( Q = \int_0^L \frac{\lambda_0 x}{L} \, dx \).
4. Perform the integration: \( Q = \frac{\lambda_0}{L} \int_0^L x \, dx \). Solve the integral \( \int_0^L x \, dx \) which results in \( \frac{L^2}{2} \).
5. Solve for \( \lambda_0 \) by equating \( Q = \frac{\lambda_0}{L} \cdot \frac{L^2}{2} \) and simplifying to find \( \lambda_0 = \frac{2Q}{L} \).

Key Concepts

Linear Charge Density

Linear charge density (λ) is defined as the amount of electric charge per unit length along a charged object. In this case, the charge distribution is nonuniform, meaning that λ varies along the length of the rod. The expression λ(x) = λ₀x/L indicates that the charge density increases linearly from the left end to the right end of the rod.

Total Charge

Total charge (Q) is the sum of all the electric charges present in a system. For the rod with a nonuniform charge distribution, the total charge can be found by integrating the linear charge density over the length of the rod. This relationship is crucial for determining λ₀, as it connects the charge distribution to the overall charge of the rod.

Integration in Physics

Integration is a fundamental mathematical tool used in physics to calculate quantities that accumulate over a continuous range. In this context, to find the total charge Q from the linear charge density λ(x), one must integrate λ(x) over the length of the rod. This process allows us to derive relationships between varying quantities and is essential for solving problems involving charge distributions.