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ChatGPT
Jul 16
Polya's law of force states that the force acting on an object is directly proportional to the rate of change of its momentum. In other words, the force applied to an object is equal to the rate of change of its momentum.
Polya's law of work states that the work done on an object is equal to the change in its kinetic energy. In other words, the work done on an object is equal to the change in its energy due to its motion.
Polya's law of work states that the work done on an object is equal to the change in its kinetic energy. In other words, the work done on an object is equal to the change in its energy due to its motion.
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Google Gemini AI
Jul 16
Polygon's Law of Force
Polygon's law of force states that the total force acting on a point charge inside a convex polygon is zero if the charge is equidistant from all vertices of the polygon. This law is a consequence of Gauss's law in electrostatics.
Polygon's Law of Work
Polygon's law of work states that the work done in moving a point charge around a closed path inside a convex polygon is zero if the charge is equidistant from all vertices of the polygon. This law is a consequence of the fact that the electric field inside a convex polygon is conservative.
Mathematical Formulation
Let's consider a point charge q located inside a convex polygon with n vertices. Let r_i be the distance between the charge and the i-th vertex. According to the law of force, the total force acting on the charge is given by:
```
F = k q ∑(i=1 to n) (1/r_i^2)
```
where k is the Coulomb constant. If the charge is equidistant from all vertices, then r_i = r for all i, and the total force becomes:
```
F = k q (n/r^2)
```
However, since n is the number of sides in a polygon, the sum of the angles in the polygon is (n-2)π. Therefore, the charge is located at the center of the polygon, where the net force is zero.
For the law of work, let's consider a closed path inside the polygon. The work done in moving the charge around this path is given by:
```
W = ∫ F ⋅ dr
```
where F is the force acting on the charge and dr is the displacement vector. Since the force is zero at any point inside the polygon, the work done in moving the charge around the closed path is also zero.
Polygon's law of force states that the total force acting on a point charge inside a convex polygon is zero if the charge is equidistant from all vertices of the polygon. This law is a consequence of Gauss's law in electrostatics.
Polygon's Law of Work
Polygon's law of work states that the work done in moving a point charge around a closed path inside a convex polygon is zero if the charge is equidistant from all vertices of the polygon. This law is a consequence of the fact that the electric field inside a convex polygon is conservative.
Mathematical Formulation
Let's consider a point charge q located inside a convex polygon with n vertices. Let r_i be the distance between the charge and the i-th vertex. According to the law of force, the total force acting on the charge is given by:
```
F = k q ∑(i=1 to n) (1/r_i^2)
```
where k is the Coulomb constant. If the charge is equidistant from all vertices, then r_i = r for all i, and the total force becomes:
```
F = k q (n/r^2)
```
However, since n is the number of sides in a polygon, the sum of the angles in the polygon is (n-2)π. Therefore, the charge is located at the center of the polygon, where the net force is zero.
For the law of work, let's consider a closed path inside the polygon. The work done in moving the charge around this path is given by:
```
W = ∫ F ⋅ dr
```
where F is the force acting on the charge and dr is the displacement vector. Since the force is zero at any point inside the polygon, the work done in moving the charge around the closed path is also zero.
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