Motion of a Block On an Inclined Plane

Projectile Motion With Air Drag Interactive Simulation

Projectile Motion — Interactive

A simple projectile motion simulation - tweak speed, angle, and see velocity vectors.

g = 9.81 m/s²
Speed: 0 m/s
Velocity: (0,0) m/s
Time: 0.00 s
Position: (0,0) m
Max Height: 0.00 m
Time of Flight: 0.00 s
Kinetic Energy: 0 J
Potential Energy: 0 J
Total Energy: 0 J
Interactive projectile motion — energy and velocity visualization.

Atwood Machine

Atwood Machine Controls

3.0 kg
1.0 kg
0.12 m
0.000 kg·m²
0.020 N·m·s
Legend — Vectors shown:
Tension, Weight.
$$\textbf{Equation for Acceleration}$$
$$ a = \frac{(m_1 - m_2)g - \dfrac{b}{R}\,\omega}{m_1 + m_2 + \dfrac{J}{R^2}} $$
Tip: change masses, pulley inertia (J) or friction (b) to see dynamics.
KAZ | Atwood Machine | Inspired by 3Blue1Brown

Interactive Simple Pendulum Simulation - A Simple, Simple Pendulum Simulator

Pendulum Controls & Energies

Energy (J)

Kinetic
0
Potential
0
Total
0
Legend — Vectors shown:
Tension, Weight, Restoring.
$$\textbf{Equation of Motion}$$
$$\frac{\mathrm{d^2} \theta}{\mathrm{d} t^2} + \frac{b}{{m}{L^2 }}\frac{\mathrm{d}\theta}{\mathrm{d} t} + \frac{g}{L}\sin\theta = 0$$
Tip: Adjust sliders to see effects. Equation governs motion.
KAZ | Inspired by 3Blue1Brown |

Invariance of Maxwell Equations Under Parity Transformation [NOT IN THE SYLLABUS. EXPLANETORY NOTE FOR PARITY VIOLATION IN BETA DECAY]

Maxwell’s Equations and Parity Invariance

One of the most beautiful features of Maxwell’s equations is their symmetry. In this post, we will prove that Maxwell’s equations are invariant under a parity transformation (spatial inversion).

1. What is a Parity Transformation?

A parity transformation (spatial inversion) is defined by:

\[ \hat{\mathbf{P}} \;\longmapsto\; \mathbf{r}' = -\mathbf{r}, \qquad t' = t \]

Transformation rules:

  • Polar vector \( \mathbf{V} \): \(\mathbf{V}'(\mathbf r',t) = -\mathbf{V}(\mathbf r,t)\)
  • Axial vector \( \mathbf{W} \): \(\mathbf{W}'(\mathbf r',t) = +\mathbf{W}(\mathbf r,t)\)
  • Scalar \( \rho \): \(\rho'(\mathbf r',t) = \rho(\mathbf r,t)\)

Gradient operator: \[ \nabla' = \frac{\partial}{\partial \mathbf r'} = -\nabla \]

In Maxwell’s theory:

  • \( \mathbf{E}, \mathbf{J} \) are polar vectors
  • \( \mathbf{B} \) is an axial vector
  • \( \rho \) is a scalar

2. Maxwell’s Equations in SI Units

\[ \begin{aligned} &(1) \quad \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \\[6pt] &(2) \quad \nabla \cdot \mathbf{B} = 0 \\[6pt] &(3) \quad \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} \\[6pt] &(4) \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned} \]

3. Applying Parity

(1) Gauss’s Law for \( \mathbf{E} \)

\[ \nabla' \cdot \mathbf{E}' = (-\nabla) \cdot (-\mathbf{E}) = \nabla \cdot \mathbf{E} \] \[ \rho'/\varepsilon_0 = \rho/\varepsilon_0 \] ✅ Invariant.

(2) Gauss’s Law for \( \mathbf{B} \)

\[ \nabla' \cdot \mathbf{B}' = (-\nabla) \cdot (+\mathbf{B}) = -\nabla \cdot \mathbf{B} \] Since RHS = 0, equation is unchanged. ✅ Invariant.

(3) Faraday’s Law

\[ \nabla' \times \mathbf{E}' = (-\nabla) \times (-\mathbf{E}) = \nabla \times \mathbf{E} \] \[ -\frac{\partial \mathbf{B}'}{\partial t} = -\frac{\partial \mathbf{B}}{\partial t} \] ✅ Invariant.

(4) Ampère–Maxwell Law

\[ \nabla' \times \mathbf{B}' = (-\nabla) \times (+\mathbf{B}) = -(\nabla \times \mathbf{B}) \] \[ \mu_0 \mathbf{J}' + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}'}{\partial t} = \mu_0 (-\mathbf{J}) + \mu_0 \varepsilon_0 \frac{\partial (-\mathbf{E})}{\partial t} \] \[ = -\left(\mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \] Both sides gain the same minus sign. ✅ Invariant.

4. Continuity Equation

\[ \frac{\partial \rho'}{\partial t} + \nabla' \cdot \mathbf{J}' = \frac{\partial \rho}{\partial t} + (-\nabla) \cdot (-\mathbf{J})\] \[ = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 \]

5. Conclusion

By treating \( \mathbf{E} \) as a polar vector, \( \mathbf{B} \) as an axial vector, and assigning the correct transformation rules to sources, we find that all four Maxwell equations remain unchanged under parity transformation.

This symmetry is one of the many reasons Maxwell’s theory is so elegant: it respects both the structure of space and the distinction between vectors and pseudovectors.


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