INT The surface charge density on an infinite charged plane is −2.0×10⁻⁶ C/m². A proton is shot straight away from the plane at 2.0×10⁶ m/s. How far does the proton travel before reaching its turning point?

Verified step by step guidance

1. Step 1: Recognize that the problem involves a charged plane creating an electric field and a proton moving in this field. The turning point occurs when the proton's initial kinetic energy is completely converted into electric potential energy.
2. Step 2: Calculate the electric field (E) created by the infinite charged plane using the formula for an infinite plane: , where $\backslash (\backslash text\{\sigma \}\backslash )\; =\; -2.0\; imes\; 10^\{-6\}\; \backslash (\backslash text\{\; C/m\}\backslash )^2$ is the surface charge density and $\backslash (\backslash text\{\epsilon \}\backslash )\_0\; =\; 8.85\; imes\; 10^\{-12\}\; \backslash (\backslash text\{\; C\}\backslash )^2/\backslash (\backslash text\{N·m\}\backslash )^2$ is the permittivity of free space.
3. Step 3: Write the energy conservation equation. The initial kinetic energy of the proton is given by , where $m\; =\; 1.67\; imes\; 10^\{-27\}\; \backslash (\backslash text\{\; kg\}\backslash )$ is the mass of the proton and $v\; =\; 2.0\; imes\; 10^6\; \backslash (\backslash text\{\; m/s\}\backslash )$ is its initial velocity. The electric potential energy at the turning point is $U\; =\; qEd$, where $q\; =\; 1.6\; imes\; 10^\{-19\}\; \backslash (\backslash text\{\; C\}\backslash )$ is the charge of the proton, $E$ is the electric field, and $d$ is the distance traveled by the proton.
4. Step 4: Set the initial kinetic energy equal to the electric potential energy at the turning point: . Solve for $d$: .
5. Step 5: Substitute the known values for $m$, $v$, $q$, and $E$ into the equation for $d$ to find the distance the proton travels before reaching its turning point.

Key Concepts

Electric Field due to a Charged Plane

An infinite charged plane creates a uniform electric field that is constant in magnitude and direction. The electric field (E) produced by a surface charge density (σ) is given by E = σ / (2ε₀), where ε₀ is the permittivity of free space. This electric field exerts a force on charged particles, such as protons, causing them to accelerate or decelerate depending on the charge and direction of the field.

Kinematics of Charged Particles

The motion of charged particles in an electric field can be analyzed using kinematic equations. When a proton is shot away from a charged plane, it experiences a force due to the electric field, which affects its velocity and position over time. The turning point occurs when the proton's velocity becomes zero, allowing us to use energy conservation or kinematic equations to determine the distance traveled before this point.

Conservation of Energy

The principle of conservation of energy states that the total mechanical energy of a system remains constant if only conservative forces are acting. In this scenario, the kinetic energy of the proton is converted into electric potential energy as it moves against the electric field. By equating the initial kinetic energy to the potential energy at the turning point, we can calculate the distance the proton travels before stopping.