A charged particle with mass m, and a charge q, is moving through a perpendicular magnetic field of strength B at a velocity,v. The particle is deflected through a curved path with a radius of 15.0 cm. If the speed of the same particle is doubled and the magnetic field is increased by a factor of 3, the particle will now be deflected through a radius of ?
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To determine the new radius of the particle's curved path, we can use the equation for the radius of curvature in a magnetic field:
r = (m * v) / (q * B)
Given that the original radius of the particle's curved path is 15.0 cm, we can substitute the known values into the equation:
15.0 cm = (m * v) / (q * B)
Now, let's consider the changes in speed and magnetic field and find the new radius of curvature.
First, we are told that the speed of the particle is doubled. Let's denote the new velocity as 2v. Substituting this into the equation, we get:
15.0 cm = (m * 2v) / (q * B)
Next, we are told that the magnetic field is increased by a factor of 3. Denoting the new magnetic field as 3B, we can substitute this into the equation:
15.0 cm = (m * 2v) / (q * 3B)
Rearranging the equation to solve for the new radius, we have:
r = (m * 2v) / (q * 3B)
r = (2mv) / (3qB)
Therefore, the new radius of curvature is given by:
r = (2mv) / (3qB)
Note: It is important to maintain consistent units throughout the calculations. If necessary, ensure that all values are in a compatible unit, such as meters, before plugging them into the equation.