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Chapter 19: Problem 27

A singly charged ion of unknown mass moves in a circle of radius $12.5 \mathrm{cm}\( in a magnetic field of \)1.2 \mathrm{T}$. The ion was accelerated through a potential difference of \(7.0 \mathrm{kV}\) before it entered the magnetic field. What is the mass of the ion?

Short Answer

Expert verified
Answer: The mass of the ion is approximately \(9.24 脳 10^{-27}\ kg\).

Step by step solution

01

Write down the formula for magnetic force

The magnetic force acting on a charged particle in a magnetic field is given by the formula: \[F_m = qvB\sin(\theta)\] where \(F_m\) = magnetic force (N) \(q\) = charge of the particle (C) \(v\) = velocity of the particle (m/s) \(B\) = magnetic field (T) \(\theta\) = angle between the velocity vector and magnetic field vector In this case, the ion is singly charged, so its charge is equal to the elementary charge (\(q = 1.6 脳 10^{-19} C\)). The ion moves in a circular path, so the angle between the velocity vector and the magnetic field vector is \(90^\circ\). Therefore, \(\sin(\theta) = 1\).
02

Write down the formula for centripetal force

The centripetal force required for the particle's circular motion is given by the formula: \[F_c = \frac{mv^2}{r}\] where \(F_c\) = centripetal force (N) \(m\) = mass of the particle (kg) \(v\) = velocity of the particle (m/s) \(r\) = radius of the circular path (m)
03

Equate magnetic force and centripetal force

Since the magnetic force is responsible for the centripetal force, we can equate the two forces: \[qvB = \frac{mv^2}{r}\]
04

Solve for velocity

Before we can find the mass, we need to determine the velocity of the ion, which can be calculated using the potential difference the ion was accelerated through: \[V = \frac{1}{2}mv^2\] Solve for \(v\): \[v = \sqrt{\frac{2qV}{m}}\]
05

Substitute the velocity expression into the magnetic force equation

Now substitute the velocity expression from Step 4 into the equation from Step 3: \[qB = \frac{m\left(\sqrt{\frac{2qV}{m}}\right)^2}{r}\]
06

Solve for mass

Now we can solve for the mass: \[m = \frac{2q^2Vr}{B^2}\]
07

Substitute the given values into the mass equation and find the mass

Substitute the given values into the mass equation: \(q = 1.6 脳 10^{-19}\ C\) \(V = 7.0 脳 10^3\ V\) \(B = 1.2\ T\) \(r = 0.125\ m\) \[m = \frac{2(1.6 脳 10^{-19}\ C)^2(7.0 脳 10^3\ V)(0.125\ m)}{(1.2\ T)^2}\] After calculating, we get: \[m \approx 9.24 脳 10^{-27} kg\] The mass of the ion is approximately \(9.24 脳 10^{-27}\ kg\).

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