A charged particle is moving in a uniform magnetic field \( \vec{B} \) with a constant speed \( v \) in a circular path of radius \( r \). Which of the following graphs represents the variation of radius of the circle, with the magnitude of magnetic field \( \vec{B} \)?
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Remember, for a charged particle moving in a magnetic field, the radius of the circular path is inversely proportional to the magnetic field strength.
To solve this problem, we need to analyze the relation between the radius of the circular path of a charged particle and the magnitude of the magnetic field in which it is moving.
The radius \( r \) of the circular path is given by the formula:
\(r = \frac{mv}{qB}\)
where:
\(m\) is the mass of the particle,
\(v\) is the speed of the particle,
\(q\) is the charge of the particle,
\(B\) is the magnitude of the magnetic field.
From the formula, it's clear that the radius \( r \) is inversely proportional to the magnetic field \( B \). Thus, as \( B \) increases, the radius \( r \) decreases following an inverse relationship.
The graph that represents an inverse relationship between \( r \) and \( B \) is a hyperbola. Among the given options, the graph in option B represents this inverse relationship:
Therefore, the correct answer is Option B.
This graph correctly shows that as the magnetic field strength increases, the radius of the path decreases, confirming the inverse proportionality.
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