CALC A 12-cm-long thin rod has the nonuniform charge density $\lambda \left(x\right)=\left(2.0\text{nC/cm}\right)e^{-\mid x\mid /\left(6.0\text{cm}\right)}$, where x is measured from the center of the rod. What is the total charge on the rod? Hint: This exercise requires an integration. Think about how to handle the absolute value sign.

Verified step by step guidance

1. Step 1: Understand the problem. The charge density λ(x) is given as a function of position along the rod, λ(x) = (2.0 nC/cm)e^(-|x|/(6.0 cm)). To find the total charge on the rod, we need to integrate λ(x) over the length of the rod, which spans from -6 cm to +6 cm (since the rod is 12 cm long and centered at x = 0).
2. Step 2: Handle the absolute value in the exponent. The absolute value |x| affects the integration because the function changes behavior at x = 0. Split the integral into two parts: one for the range from -6 cm to 0 cm and another for the range from 0 cm to +6 cm. This allows us to remove the absolute value by considering the sign of x in each region.
3. Step 3: Write the integral for the total charge. The total charge Q is given by Q = ∫[−6 cm to +6 cm] λ(x) dx. After splitting the integral, this becomes Q = ∫[−6 cm to 0 cm] (2.0 nC/cm)e^(x/(6.0 cm)) dx + ∫[0 cm to +6 cm] (2.0 nC/cm)e^(−x/(6.0 cm)) dx.
4. Step 4: Solve each integral separately. For the first integral (from -6 cm to 0 cm), substitute λ(x) = (2.0 nC/cm)e^(x/(6.0 cm)). For the second integral (from 0 cm to +6 cm), substitute λ(x) = (2.0 nC/cm)e^(−x/(6.0 cm)). Use the standard formula for integrating exponential functions: ∫e^(kx) dx = (1/k)e^(kx), where k is the constant in the exponent.
5. Step 5: Combine the results of the two integrals. After evaluating both integrals, add their results to find the total charge Q on the rod. Ensure that the units are consistent throughout the calculation (e.g., converting nC/cm to nC if necessary).

Key Concepts

Charge Density

Charge density refers to the amount of electric charge per unit length, area, or volume. In this case, the rod has a linear charge density λ(x), which varies along its length. Understanding how charge density is defined and how it can change with position is crucial for calculating the total charge on the rod.

Integration

Integration is a mathematical process used to find the total accumulation of a quantity, such as charge, over a specified interval. In this problem, integration is necessary to sum the contributions of the varying charge density along the length of the rod. Recognizing how to set up and evaluate the integral is essential for solving the problem.

Absolute Value

The absolute value function, denoted as |x|, represents the distance of a number from zero, disregarding its sign. In this context, it is important to handle the absolute value correctly when integrating the charge density, as it affects the limits of integration and the expression for λ(x) based on the position along the rod.