two particle with equal mass, one carrying a charge q and the other -2q each enter at right angles to a uniform magnetic field with the same velocity. which is deflected into a circular path with the larger radius?
To determine which particle is deflected into the larger circular path, we need to use the principles of magnetic forces and circular motion.
First, let's consider the equation for the force experienced by a charged particle moving through a magnetic field:
F = q * v * B * sin(θ)
Where:
- F is the magnetic force acting on the particle,
- q is the charge of the particle,
- v is the velocity of the particle,
- B is the magnetic field strength, and
- θ is the angle between the velocity vector and the magnetic field vector.
Since both particles have the same mass, same velocity, and enter at right angles to the magnetic field, we can simplify the equation to:
F = q * v * B
From this equation, we can observe that the force experienced by a particle is directly proportional to its charge.
Now, for circular motion, we know that the centripetal force required to keep an object in a circular path is given by:
F = m * v^2 / r
Where:
- F is the centripetal force,
- m is the mass of the particle,
- v is the velocity of the particle, and
- r is the radius of the circular path.
Since both particles have the same mass and velocity, the centripetal force is the same for both. Therefore, we can set up an equation as follows:
m * v^2 / r = q * v * B
We cancel out the common terms:
v / r = q * B
Now, let's analyze the equation. As we can see, the radius of the circular path, r, is inversely proportional to the charge, q. Thus, the particle with the larger charge will have the smaller radius, and vice versa. Therefore, the particle carrying a charge of -2q will be deflected into a circular path with a larger radius compared to the particle carrying a charge of q.
In conclusion, the particle with the charge q is deflected into a circular path with the larger radius.