A particle with a charge of -60.0 nC is placed at the center of a nonconducting spherical shell of inner radius 20.0 cm and outer radius 33.0 cm. The spherical shell carries charge with a uniform density of -2.02 µC/m3. A proton moves in a circular orbit just outside the spherical shell. Calculate the speed of the proton.
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To calculate the speed of the proton, we can use the principles of electrostatics and circular motion. Here's how you can approach this problem:
1. Determine the electric field inside the nonconducting spherical shell:
- The electric field inside a uniformly charged spherical shell is zero. This is because the electric field exerted by the positively charged inner surface cancels out the electric field exerted by the negatively charged outer surface.
2. Determine the electric field just outside the nonconducting spherical shell:
- The electric field just outside a uniformly charged spherical shell is given by the equation E = Q / (4πε₀r²), where Q is the total charge enclosed by the shell, ε₀ is the vacuum permittivity, and r is the distance from the center of the shell.
- In this case, the total charge enclosed by the shell is the charge density multiplied by the volume of the shell: Q = ρ * V.
- Calculate the volume of the shell using V = (4/3)π(r_outer³ - r_inner³).
- Substitute the given charge density (-2.02 µC/m³) and the given radii (20.0 cm and 33.0 cm) to find the total charge enclosed.
- Substitute the values into the electric field equation to find the electric field just outside the shell.
3. Determine the force exerted on the proton by the electric field:
- The force exerted on a charged particle moving in an electric field is given by the equation F = qE, where q is the charge of the particle and E is the electric field.
- In this case, the charge of the proton is the elementary charge e, q = e.
- Substitute the value of the electric field just outside the shell and the charge of the proton to find the force exerted on the proton.
4. Determine the centripetal force required to keep the proton in circular motion:
- Since the proton is moving in a circular orbit, it experiences a centripetal force given by the equation F_c = m*v²/r, where m is the mass of the proton, v is its velocity, and r is the radius of the circular orbit.
- Rearrange the equation to solve for v: v = √(F_c * r / m).
5. Find the velocity of the proton:
- Substitute the force exerted on the proton by the electric field and the given radius (distance from the shell) into the centripetal force equation to find the velocity of the proton.
By following these steps, you should be able to calculate the speed of the proton.