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How Can Differential Equations Help Solve Homework Problems?
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Differential equations can be instrumental in solving various homework problems by modeling real-world scenarios and providing a framework for finding solutions. The forum emphasizes that direct answers to homework questions are not permitted; instead, participants are encouraged to engage in problem-solving discussions. The effectiveness of using differential equations lies in their ability to describe dynamic systems and changes over time. Participants are invited to share their thoughts on how to approach specific problems using these mathematical tools. Engaging with differential equations can enhance understanding and lead to successful problem resolution.
At first, I derived that:
$$\nabla \frac 1{\mu}=-\frac 1{{\mu}^3}\left((1-\beta^2)+\frac{\dot{\vec\beta}\cdot\vec R}c\right)\vec R$$
(dot means differentiation with respect to ##t'##).
I assume this result is true because it gives valid result for magnetic field.
To find electric field one should also derive partial derivative of ##\vec A## with respect to ##t##. I've used chain rule, substituted ##\vec A## and used derivative of product formula.
$$\frac {\partial \vec A}{\partial t}=\frac...
Hi, I'm stuck at this question, please help.
Attempt to the Conducting Sphere and Dipole Problem
(a) Electric Field and Potential at O due to Induced Charges
$$V_O = 0$$
This potential is the sum of the potentials due to the real charges (##+q, -q##) and the induced charges on the sphere.
$$V_O = V_{\text{real}} + V_{\text{induced}} = 0$$
- Electric Field at O, ##\vec{E}_O##: Since point O is inside a conductor in electrostatic equilibrium, the electric field there must be zero...