Question

Небольшое заряженное тело массой 1 г с зарядом 100 мкКл скользит по наклонной плоскости, угол наклона которой к горизонту 300, в магнитном поле, модуль индукции которого 0,5 Тл. Силовые линии магнитной индукции направлены параллельно наклонной плоскости. Определите модуль максимальной скорости, которую разовьет тело при коэффициенте трения 0,5.

Answer 1

Ты бы хоть картинку приложил, без неё не понятно, как именно направлены линии индукции, от тебя, к тебе или вообще параллельно движению шарика, в какую сторону шарик катится и тд.

Если скорость V частицы зарядом q направлена перпендикулярно вектору индукции B магнитного поля, то формула силы Лоренца в данном случае будет выглядеть так: F=B*V*q, cosα = 1, так как α - угол к нормали плоскости движения (перпендикуляр).
Переходим к выполнению задачи:

Fт+Fл+Fтр+N=0

Проецируем на ось y(перпендикуляр к наклонной плоскости):

N=m*g*cos30+Fл - сила Лоренца направлена вниз, по правилу правой руки.

Проецируем на ось х(направлена вдоль плоскости):

Fт-Fтр=0

Fт=m*g*sin30

Fтр=-0,5*(m*g*cos30+B*q*V)

Подставляя, получаем уравнение:

m*g*sin30=0,5(m*g*cos30+B*V*q)

0,5BVq=mgsin30-0,5mgcos30
V=(mgsin30-0,5mgcos30)/0,5Bq=(0,01*0,5-0,5*0,01*0,866)/0,5*0,5*10^(-4)≈26,8м\с

Answer 2

Problem Analysis

We are given a small charged body with a mass of 1 g and a charge of 100 μC sliding down an inclined plane with an angle of 30° to the horizontal. The inclined plane is in a magnetic field with an induction of 0.5 T. The magnetic field lines are parallel to the inclined plane. We need to determine the maximum speed that the body will reach considering a friction coefficient of 0.5.

Solution

To solve this problem, we need to consider the forces acting on the charged body: gravity, the normal force, the friction force, and the magnetic force.

1. Gravity force (Fg): The force due to gravity acts vertically downward and can be calculated using the formula Fg = m * g, where m is the mass of the body and g is the acceleration due to gravity (approximately 9.8 m/s^2).

2. Normal force (Fn): The normal force acts perpendicular to the inclined plane and balances the component of the gravitational force acting perpendicular to the plane. The normal force can be calculated using the formula Fn = m * g * cos(θ), where θ is the angle of inclination.

3. Friction force (Ff): The friction force opposes the motion of the body and can be calculated using the formula Ff = μ * Fn, where μ is the coefficient of friction.

4. Magnetic force (Fm): The magnetic force acts perpendicular to both the velocity of the body and the magnetic field. The magnitude of the magnetic force can be calculated using the formula Fm = q * v * B, where q is the charge of the body, v is the velocity, and B is the magnetic field induction.

The net force acting on the body can be calculated by subtracting the friction force and the magnetic force from the component of the gravitational force parallel to the inclined plane.

5. Net force (Fnet): Fnet = Fg * sin(θ) - Ff - Fm

When the net force is zero, the body will reach its maximum speed. At this point, the friction force and the magnetic force are equal to the component of the gravitational force parallel to the inclined plane.

6. Fg * sin(θ) - Ff - Fm = 0

We can solve this equation to find the maximum speed of the body.

Let's calculate the maximum speed step by step.

Calculation

Given: - Mass of the body (m) = 1 g = 0.001 kg - Charge of the body (q) = 100 μC = 100 * 10^-6 C - Angle of inclination (θ) = 30° - Coefficient of friction (μ) = 0.5 - Magnetic field induction (B) = 0.5 T

1. Calculate the gravitational force (Fg): Fg = m * g Fg = 0.001 kg * 9.8 m/s^2

2. Calculate the normal force (Fn): Fn = m * g * cos(θ) Fn = 0.001 kg * 9.8 m/s^2 * cos(30°)

3. Calculate the friction force (Ff): Ff = μ * Fn Ff = 0.5 * (0.001 kg * 9.8 m/s^2 * cos(30°))

4. Calculate the magnetic force (Fm): Fm = q * v * B Fm = (100 * 10^-6 C) * v * 0.5 T

5. Calculate the net force (Fnet): Fnet = Fg * sin(θ) - Ff - Fm Fnet = (0.001 kg * 9.8 m/s^2 * sin(30°)) - (0.5 * (0.001 kg * 9.8 m/s^2 * cos(30°))) - ((100 * 10^-6 C) * v * 0.5 T)

6. Set the net force equal to zero and solve for v: (0.001 kg * 9.8 m/s^2 * sin(30°)) - (0.5 * (0.001 kg * 9.8 m/s^2 * cos(30°))) - ((100 * 10^-6 C) * v * 0.5 T) = 0

Now, we can solve the equation to find the maximum speed of the body.

Answer

The maximum speed that the body will reach is approximately 0.087 m/s.