Question

An electron is moving at \(v=6.00 \cdot 10^{7} \mathrm{~m} / \mathrm{s}\) perpendicular to the Earth's magnetic field. If the field strength is \(0.500 \cdot 10^{-4} \mathrm{~T}\), what is the radius of the electron's circular path?

Short answer

Answer: The radius of the electron's circular path is approximately \(3.41 \cdot 10^{-6} \mathrm{~m}\).

Step-by-step solution

Magnetic force on the electron due to the magnetic field

The magnetic force on a charged particle moving in a magnetic field can be calculated using the formula: $$ F = qvB\sin \theta $$ where \(F\) is the magnetic force, \(q\) is the charge of the particle, \(v\) is its velocity, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the particle's velocity and the magnetic field. Since the electron is moving perpendicular to the Earth's magnetic field, \(\theta = 90^\circ\), and \(\sin \theta = 1\). The charge of an electron is \(q = -1.60 \cdot 10^{-19} \mathrm{~C}\), so: $$ F = (-1.60 \cdot 10^{-19} \mathrm{~C})(6.00 \cdot 10^{7} \mathrm{~m/s})(0.500 \cdot 10^{-4} \mathrm{~T})(1) $$

Calculate the magnetic force (F)

Computing the values above, we get: $$ F = -4.80 \cdot 10^{-16} \mathrm{~N} $$ Since force is always a positive value in this context, we take the absolute value: $$ F = 4.80 \cdot 10^{-16} \mathrm{~N} $$

Centripetal force in circular motion

The centripetal force acting on the electron can be described as \(F_c = m_ev^2/r\), where \(F_c\) is the centripetal force, \(m_e\) is the electron's mass, and \(r\) is the radius of the circular path. In this case, the magnetic force is the centripetal force acting on the electron. Thus: $$ F_c = F = m_ev^2/r $$ We can rearrange the formula to solve for \(r\): $$ r = \frac{m_ev^2}{F} $$ The mass of an electron is \(m_e = 9.11 \cdot 10^{-31} \mathrm{~kg}\), so:

Calculate the radius of the electron's circular path (r)

Plugging in the values, we get: $$ r = \frac{(9.11 \cdot 10^{-31} \mathrm{~kg})(6.00 \cdot 10^{7} \mathrm{~m/s})^2}{4.80 \cdot 10^{-16} \mathrm{~N}} $$ Calculating this expression, we find the radius of the electron's circular path to be: $$ r \approx 3.41 \cdot 10^{-6} \mathrm{~m} $$

Key concepts

Magnetic Force

When an electron or any charged particle moves through a magnetic field, it experiences a magnetic force. This force is the result of the interaction between the charge's motion and the magnetic field. To determine this force, one can use the formula:
\[ F = qvB\sin(\theta) \]
where

• \( F \) is the magnetic force exerted on the particle,
• \( q \) is the charge of the particle,
• \( v \) is the velocity of the particle,
• \( B \) is the strength of the magnetic field,
• and \( \theta \) is the angle between the velocity vector and the magnetic field.

For an electron, which carries a negative charge, the direction of the force is given by the right-hand rule and is perpendicular to both its velocity and the magnetic field. If the electron travels perpendicular to the field, as in the original exercise, \( \theta = 90^\circ \), making \( \sin(\theta) = 1 \). Thus, the magnitude of the force on an electron can be calculated without considering the sine component, simplifying the computation.

Centripetal Force

The centripetal force is an inward force that is required to keep an object moving in a circular path. It's directed towards the center of the circle and is responsible for the changes in the direction of velocity of the object, without changing its speed.
For an electron moving in a circular path under the influence of a magnetic field, the magnetic force acts as the centripetal force. The formula to express this relationship is:
\[ F_c = \frac{m_ev^2}{r} \]
where

• \( F_c \) is the centripetal force,
• \( m_e \) is the mass of the electron,
• \( v \) is the velocity of the electron,
• and \( r \) is the radius of the circular path.

The value of the magnetic force calculated from the earlier formula can be set equal to the centripetal force to solve for the electron's path radius.

Radius of Circular Path

The radius of a circular path in this context is the distance from the center of the circle to the path of the electron. It's a key component in understanding the electron's motion and is determined by the balance of forces acting on the electron. Rearranging the centripetal force formula yields:
\[ r = \frac{m_ev^2}{F} \]
where the mass \( m_e \) of the electron and its velocity \( v \) are constants, while \( F \) is the magnetic force resulting from the electron's motion in the magnetic field. When these values are known, as in the exercise provided, the radius of the electron's circular path can be calculated with ease. This radius dictates how tightly the electron will curve as it travels through the magnetic field.

Circular Motion in Magnetic Field

Circular motion of an electron in a magnetic field demonstrates a fundamental electromagnetic interaction. As the electron moves perpendicular to the field lines, it is deflected by the magnetic force acting as a centripetal force. This causes the electron to move in a circular path with a radius determined by its speed, the magnetic field's strength, and the electron's mass.
This circular motion can be observed in many applications including cyclotrons and mass spectrometers, devices that utilize magnetic fields to control the motion of charged particles. Understanding the balance of forces and the resulting circular motion is crucial in physics and engineering for harnessing the behavior of charged particles in a controlled manner.