\(\What\) is the value of q/m for a particle that moves in a circle of radius 8.0 mm in a 0.46-T magnetic field if a crossed 320-V/m electric field will make the path straight?

검증된 단계별 안내

Understand the problem: The particle is moving in a circular path due to the magnetic field, and the electric field is applied to make the path straight. This means the electric force and magnetic force must balance each other. Use this condition to find the charge-to-mass ratio (q/m).

Write the force balance equation: The electric force \( F_E \) is given by \( F_E = qE \), and the magnetic force \( F_B \) is given by \( F_B = qvB \). For the path to be straight, \( F_E = F_B \), so \( qE = qvB \). Simplify to get \( v = \frac{E}{B} \), where \( v \) is the velocity of the particle.

Relate the circular motion to the magnetic force: In the absence of the electric field, the particle moves in a circle due to the magnetic force. The centripetal force is provided by the magnetic force, so \( F_B = \frac{mv^2}{r} \). Substituting \( F_B = qvB \), we get \( qvB = \frac{mv^2}{r} \). Simplify to find \( \frac{q}{m} = \frac{v}{Br} \).

Substitute \( v = \frac{E}{B} \) into \( \frac{q}{m} = \frac{v}{Br} \): Replace \( v \) with \( \frac{E}{B} \) to get \( \frac{q}{m} = \frac{\frac{E}{B}}{Br} \). Simplify this to \( \frac{q}{m} = \frac{E}{B^2r} \).

Substitute the given values into the formula: Use \( E = 320 \ \text{V/m} \), \( B = 0.46 \ \text{T} \), and \( r = 8.0 \ \text{mm} = 8.0 \times 10^{-3} \ \text{m} \) in the equation \( \frac{q}{m} = \frac{E}{B^2r} \) to calculate the charge-to-mass ratio.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through electric and magnetic fields. It is given by the equation F = q(E + v × B), where F is the force, q is the charge, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This concept is crucial for understanding how charged particles behave in electromagnetic fields.

Cyclotron Motion

Cyclotron motion refers to the circular path that a charged particle follows when it moves perpendicular to a uniform magnetic field. The radius of this path is determined by the particle's velocity, charge, and the strength of the magnetic field. This concept is essential for analyzing the motion of the particle in the given magnetic field and understanding how the electric field can alter this motion.

Charge-to-Mass Ratio (q/m)

The charge-to-mass ratio (q/m) is a measure of how much charge a particle has relative to its mass. It plays a significant role in determining the radius of the circular path of a charged particle in a magnetic field. By calculating q/m, one can predict the behavior of the particle under the influence of electric and magnetic fields, which is key to solving the problem presented.