Gauss’s Law in One Dimension and in Three Dimensions

Recall the following form of Gauss’s Law from this post: $ \frac{\partial}{\partial x} E_x +  \frac{\partial}{\partial y} E_y + \frac{\partial}{\partial z} E_z = \frac{\rho}{\epsilon_0 \epsilon_r} $ One Dimension If a problem is “one-dimensional” along the, say, $x$-axis, one has $ \frac{\partial}{\partial x} E_x  = \frac{\rho}{\epsilon_0 \epsilon_r} $ In words: the partial derivative with respect to $x$…

Gauss’s Law in Differential Form and Cartesian Coordinates

Gauss’s Law in Differential FormIn differential form, Gauss’s Law is $ \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0 \epsilon_r} $ The next part of this post attempts to demystify this law a bit. Description of SymbolsThe $\vec{\nabla}$ on the left is called ‘del’, and it can be written in terms of Cartesian unit vectors and partial derivatives…

Challenging Math and Physics Topics from High School and College

Hi All, I am planning on writing blog posts focused on math and physics topics, specifically those that students find challenging in high school and college. So, if you could let me know what you found difficult during your college or high school experience, by emailing me or by leaving a comment on this post,…

Pi

Pi, or $\pi$, is commonly used in the context of circles in geometry. In particular, the circumference $C$ of a circle that has a diameter $d$ is $C = \pi d$. Also, the area $A$ of a circle with radius $r$ is $ A = \pi r^2$. Note that $2r = d$. This is how…

Prime Number

What is a prime number? A prime number cannot be divided by another number, with the exception of the number 1 and the number itself. Some prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23. Prime numbers are involved in encryption in today’s technology. Some people think that prime numbers are…

Pythagoras Theorem

What is the Pythagoras theorem? It is $ a^2 + b^2 = c^2$. Here, $a$, $b$ and $c$ are the lengths of a right triangle. Note that $c$ is the hypotenuse of the right triangle. A right triangle has an angle of 90 degrees.

Bubble Sort Algorithm in Python

In this post, I provide a code that performs a bubble sort in python. Before using, I recommend checking the indentations, as this code was written in a standard text-editor. This python bubble-sort algorithm should be fine, though. This python algorithm was written using the example output in Reference 1. One can compare the output of…

Conservation of Momentum

If an external force is not applied to a system, then momentum is conserved. For example, consider one dimensional motion, and suppose there are two objects $A$ and $B$ with masses $m_A$ and $m_B$, respectively. If the initial velocity of object $A$ is $v_{A_i}$ and the initial velocity of object $B$ is $v_{B_i}$, then $m_A…

Newton’s Second Law

Newton’s Second Law is symbolically written as $ \vec{F}_{net} = m \vec{a}$. Here, $\vec{F}_{net}$ is the net force on an object with nonzero mass $m$. If the object has a mass of zero, such as a photon, Newton’s second law indicates that the net force on the object would be the zero vector. If the…

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