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An electric field is a region around a charged particle where it exerts a force on other charges. Imagine it as an invisible force field around the charged object. This field is represented by the symbol \( E \) and measured in volts per meter (\( \mathrm{V/m} \)).Electric fields are essential in velocity selectors, enabling control over ion motion. In the context of the velocity selector in the given problem, the electric field is responsible for exerting a force on the ions. The force \( F_e \) on a charge \( q \) in an electric field \( E \) is given by\[ F_e = qE \]For a singly ionized atom, where one electron is removed, the charge \( q \) is equivalent to the charge of a single electron, \( 1.6 \times 10^{-19} \, \mathrm{C} \). This field ensures that only ions with the desired velocity pass through, ultimately becoming a crucial part of the velocity selector's functionality.

A magnetic field surrounds magnets and current-carrying wires and applies a force on moving charges. Represented by \( B \), it is measured in teslas (\( \mathrm{T} \)).In a velocity selector, the magnetic field is perpendicular to the electric field. Here, its crucial role is to exert a magnetic force \( F_m \) on the moving ions. This force can be calculated using:\[ F_m = qvB \]Where \( q \) is the charge, \( v \) is the velocity of ions, and \( B \) is the magnetic field strength. In the given exercise, the initial magnetic field \( B_1 \) is \( 0.0315 \ \, \mathrm{T} \), perfectly aligned to cancel out the electric force.When the electric and magnetic forces are equal, ions travel straight through the selector at a constant velocity \( v \). This condition lets us derive the equation for velocity \( v = \frac{E}{B_1} \), which is an essential part of the solution. Hence, magnetic fields are key to ensuring only ions with a specific velocity emerge from the selector.

Centripetal force is what keeps an object moving in a circular path, constantly pulling it toward the center of the circle. In uniform circular motion, it is essential to balance all external forces acting to maintain the circular path.In the exercise, ions enter a uniform magnetic field \( B_2 = 0.0175 \, \mathrm{T} \) after emerging from the velocity selector. The magnetic force now becomes the centripetal force required for circular motion:\[ F_{ ext{centripetal}} = \frac{mv^2}{r} \]Here, \( m \) is the mass, \( v \) is the velocity (determined earlier as \( 4920.63 \, \mathrm{m/s} \)), and \( r \) is the radius (given as \( 0.175 \, \mathrm{m} \)). This equation helps us find the mass of the ions by equating the centripetal force with the magnetic force \( qvB_2 \). Ultimately, centripetal force dictates the path and properties of the ions within the magnetic field.

Ion acceleration refers to the process by which ions gain velocity due to electric or magnetic fields. Acceleration is crucial for ensuring ions achieve the selected velocity.In this context, ions start from a state where they are singly ionized, meaning they are initially accelerated by an electric field. This process ensures they reach sustained velocities needed to pass through the velocity selector.Once accelerated, ions enter the selector at a velocity determined by the ratio of the electric field \( E \) and magnetic field \( B_1 \):\[ v = \frac{E}{B_1} \]This relationship highlights the delicate balance between electric and magnetic fields, which ensures only ions with the matched velocity pass through without deflection. Ion acceleration provides the crucial initial energy boost allowing the velocity selector to perform its filtering task.