91Ó°ÊÓ

Gauss's Law is a fundamental principle in electromagnetism, crucial for understanding electric fields generated by various charge distributions. The law is expressed as: ewline ewline ewline \[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0} \] ewline ewline This equation states that the total electric flux through a closed surface is equal to the enclosed charge ( \(Q_{enc}\)) divided by the permittivity of free space ( \(\varepsilon_0\)). Essentially, Gauss's Law connects the electric field ( \(\mathbf{E}\)) with the charge inside the surface. In the context of a long, straight wire with a linear charge density, we use a cylindrical Gaussian surface co-axial with the wire. The symmetry of the setup ensures that the electric field at any point on the cylindrical surface is the same. This makes calculations straightforward and manageable. Gauss's Law simplifies the process by taking advantage of these symmetries. Applying Gauss's Law, we find the electric field \(E\) due to the linear charge density \(\lambda\): \[ \displaystyle E = \frac{\lambda}{2 \pi \varepsilon_0 r} \] , where \(r\) is the radial distance from the wire. This formula is pivotal for further calculations in this exercise.

Linear charge density ( \(\lambda\)) describes the amount of electric charge per unit length along a line, such as a long, thin wire. It is measured in coulombs per meter ( \(C/m\)). For instance, a linear charge density of \( -3.6 \text{ nC}/ \text{ m} \) signifies that each meter of the wire holds \( -3.6 \times 10^{-9} \text{ C}\) of charge. Linear charge density is essential for calculating the electric field around charged conductors. For a straight wire, it simplifies the complex electric field equations by using per-length charge data. In our problem, the given linear charge density allows us to use a known formula to find the electric field: \[ \displaystyle E = \frac{\lambda}{2 \pi \varepsilon_0 r} \] This step is crucial in understanding how the charge distribution affects the surrounding electric field. Additionally, knowing the linear charge density helps in designing setups to counterbalance electric fields, as is done with the cylindrical surface in this exercise. By adjusting the surface charge density on this cylinder, we aim to nullify the wire's electric field, achieving electrostatic equilibrium.

Surface charge density ( \(\sigma\)) represents the amount of charge per unit area on a surface. It is measured in coulombs per square meter ( \(C/m^2\)). For instance, a surface charge density of \(3.77 \times 10^{-1} \text{nC}/\text{m}^2 \) means that every square meter of the surface has \(3.77 \times 10^{-10} \text{C} \) of charge. In this exercise, surface charge density was crucial to balance the electric field from a negatively charged wire with a line charge density. The cylinder around the wire was given a positive charge such that its electric field cancels the wire's field. The electric field \(E_c\) due to the cylinder is: \[ E_c = \frac{\sigma}{\varepsilon_0} \]. This means the field is directly proportional to the surface charge density. To find the needed \(\sigma\) to nullify the external electric field, we set the cylinder's electric field equal and opposite to that of the wire. Here’s the formula used: \( E = E_c \); simplifying we got: \[ \sigma = -3.6 \times 10^{-9} \times 2 \pi \times 0.015 \approx 3.77 \times 10^{-1} \text{nC}/\text{m}^2 \]. By balancing these fields, we ensure that the net external electric field becomes zero, demonstrating the importance of understanding and employing surface charge density in electrostatic problems.