Q1. Attempt all parts of question 1.
- (a)) The height of a hill is given by z = 100 – x²y². Calculate the maximum rate of change (also called the steepest ascent) in the height of the hill at the point (2,1). What is its direction? (150 words)
- (b)) Consider a rigid body rotating about a fixed axis with a constant angular velocity ω, directed along the axis of rotation. The velocity of a particle on the rigid body is ω × r. Calculate ∇ × v. (150 words)
- (c)) Calculate the net work done by a force F = (6x – 2y)i + (x²)j in taking a particle along the straight line between the points (5, −3) to (0,0) and then along the straight line between the points (0,0) to (5,3). Is the force conservative? Explain. (300 words)
- (d)) Calculate the flux of a vector field F = 2xî + y ĵ+3zk through the surface of a cube defined by 0 ≤ x ≤ 2; 0 ≤ y ≤ 2 ; 0≤z≤2. (150 words)
Key Points:
- Gradient's magnitude gives maximum rate of change, its direction points steepest ascent.
- Curl of velocity for rigid body rotation is twice the angular velocity (∇ × v = 2ω).
- Work done by a non-conservative force depends on the path taken.
- A force is conservative if its curl (∇ × F) is identically zero.
Answer: This response addresses all four sub-questions from BPHCT-133 - Electricity and Magnetism, covering vector calculus concepts essential for understanding physical phenomena. Each part provides a detailed solution, demonstrating the application of gradient, curl, line integrals, and divergence theorem, which are foundational tools in electromagnetism and other areas of physics.