Numerical Problem Set 9 Physics 204B November 20, 2015 Paul ✓ Solved

Numerical Problem Set 9physics 204bnovember 20 2015 Paul Arpin1 Elect

Write a program that calculates the electric field at any point in space due to a uniformly charged ring centered on the origin in the x-y plane. Assume the total charge Q on the ring, its radius R, and the position where the electric field is measured given by the vector~rmeasure. Distribute the charge over the ring into N segments, calculate the contribution to the electric field from each segment as a point charge, and sum these contributions to find the total electric field. Use the Coulomb constant k for calculations. Also, verify the calculation against the known analytical solution along the axis of the ring at a height z = 10cm, and determine the convergence of your numerical solution with increasing N.

Sample Paper For Above instruction

The problem of calculating the electric field due to a uniformly charged ring is a classic application of Coulomb's law and vector calculus in electrostatics. When dealing with continuous charge distributions, discretization techniques such as dividing the ring into N segments enable numerical approximation of the electric field at a specified point in space. This method involves computing the electric field contribution of each charge element, modeled as a point charge, and summing these contributions vectorially to obtain the total electric field.

Geometric Setup and Assumptions: The ring is centered at the origin in the xy-plane with radius R. The total charge distributed uniformly over the ring is Q, which is divided into N segments, each carrying a charge q. The point where the electric field is to be measured, specified by the vector~rmeasure, is located at a position in space relative to the origin. This setup allows us to compute the electric field at any specified position, such as along the axis at height z, by summing the contributions from each segment.

Charge Distribution and Discretization: To discretize the ring, the ring's circumference is divided evenly into N segments, each subtending an angle Δθ = 2π/N. The position of each segment's charge can be determined using parametric equations for a circle: x_i = R cosθ_i, y_i = R sinθ_i, where θ_i = iΔθ for i in 0,...,N-1. Each charge segment q is calculated using the linear charge density λ, given by λ = Q / (2π R). The charge on each segment then becomes q = λ R Δθ.

Calculating the Electric Field Contribution: For each segment, the position vector is known. The vector~r from the segment's charge to the measurement point is obtained by subtracting the position vector of the charge from the measurement point. The magnitude of~r is |~r|. The unit vector~r̂ points from the charge to the measurement point, calculated as ~r / |~r|. The contribution to the electric field from a single segment is then: ∆~E = (k q / |~r|^2) ~r̂, where k is Coulomb's constant (8.99 x 10^9 Nm^2/C^2). Summing all these vector contributions yields the total electric field at the measurement point.

Implementation and Verification: Implementing this algorithm in Python involves setting up a loop over the N segments, computing the positions and contributions, and summing the electric fields. To verify the correctness, the calculated electric field along the z-axis at height z=10cm is compared to the analytical solution: E_z = (1 / (4πε_0)) * Q z / (R^2 + z^2)^(3/2). As N increases, the numerical solution should converge to the analytical value. The convergence rate can be observed by increasing N until the difference between numerical and analytical solutions is within 3 significant figures, typically requiring N to approach a certain threshold.

Results and Analysis: For small N, the calculated electric field tends to deviate from the analytical value, typically overestimating or underestimating depending on discretization errors. As N increases, the approximation improves, and the numerical results converge, demonstrating the efficacy of the discretization approach. The critical N for convergence can be identified by observing when the difference falls within the desired accuracy. This approach exemplifies the power of numerical methods in solving electrostatics problems where analytical solutions are complex or unwieldy.

In summary, by discretizing a continuous charge distribution on a ring and summing contributions from point charges, this program provides an effective way to compute electric fields numerically. The method aligns with analytical solutions in simple symmetric cases, validating the approach and illustrating the convergence properties of numerical integration in physics.

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