Practice Laplace Equation (Steady State Heat) - 15.2.C | 15. Fourier Series Solutions to PDEs | Mathematics - iii (Differential Calculus) - Vol 2
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15.2.C - Laplace Equation (Steady State Heat)

Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

What does the Laplace equation model in the context of heat conduction?

💡 Hint: Think about how heat flow reaches equilibrium.

Question 2

Easy

Define what boundary conditions are in solving PDEs.

💡 Hint: Consider what happens at the physical limits.

Practice 4 more questions and get performance evaluation

Interactive Quizzes

Engage in quick quizzes to reinforce what you've learned and check your comprehension.

Question 1

What form does the Laplace equation take in a 2D scenario?

  • \\( \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 0 \\)
  • \\( \\frac{\\partial u}{\\partial x} + \\frac{\\partial u}{\\partial y} = 0 \\)
  • \\( u = ax + by \\)

💡 Hint: Think about the heat conduction behavior in multiple dimensions.

Question 2

True or False: The Laplace equation can be used for non-steady-state heat conduction.

  • True
  • False

💡 Hint: Consider what steady-state means in the context of heat.

Solve 2 more questions and get performance evaluation

Challenge Problems

Push your limits with challenges.

Question 1

Given a rectangular metal plate with Dirichlet boundary conditions of 100 degrees Celsius on one edge and 0 degrees on all other edges, derive the steady-state temperature distribution.

💡 Hint: Draw the plate and system to visualize the application of boundary conditions.

Question 2

A cylindrical wire with radius R is kept at temperature T0 on the surface and cooled at the edges. Formulate the steady-state temperature profile using Laplace's Equation in cylindrical coordinates.

💡 Hint: Transform the Laplace equation correctly for cylindrical coordinates.

Challenge and get performance evaluation