Problem 29 - University Physics with Modern Physics

Question

Charge \(Q =\) 5.00 mC is distributed uniformly over the volume of an insulating sphere that has radius \(R =\) 12.0 cm. A small sphere with charge \(q = +\)3.00 \(\mu\)C and mass 6.00 \(\times 10^{-5}\) kg is projected toward the center of the large sphere from an initial large distance. The large sphere is held at a fixed position and the small sphere can be treated as a point charge. What minimum speed must the small sphere have in order to come within 8.00 cm of the surface of the large sphere?

Step-by-Step Solution

Answer

The minimum speed required is determined by applying conservation of energy using the potential inside the sphere.

Step 1: Understand the System

We have a large insulating sphere with a uniform charge distribution and a smaller point charge that is projected toward it. We need to calculate the minimum speed of the smaller sphere so that it approaches the large sphere up to 8.00 cm from its surface.

Step 2: Determine Distance of Closest Approach

The radius of the large sphere is given as 12.0 cm, so its surface is at 12 cm. The small sphere is projected to come within 8.00 cm of the surface, meaning the closest approach distance to the center of the large sphere is 12.0 cm - 8.0 cm = 4.0 cm.

Step 3: Apply Energy Conservation Principle

Initially, the small charge is at a distance, so its potential energy is effectively zero. We will use conservation of mechanical energy, where initial kinetic energy equals the sum of kinetic and potential energy at the closest approach point.

Step 4: Compute Electric Field and Potential at Closest Approach

The electric field inside a uniformly charged sphere is given by \( E = \frac{1}{4\pi\varepsilon_0}\cdot\frac{Qr}{R^3}\) where \( r \) is the distance from the center. The potential at radius \( r \) is \( V = \frac{1}{4\pi\varepsilon_0}\cdot\left( \frac{3Q}{2R} - \frac{Qr^2}{2R^3} \right) \). Evaluate these at \( r = 4 \) cm.

Step 5: Calculate Initial Speed Using Conservation of Energy

The kinetic energy needed for the small sphere to reach 4.0 cm from the center of the large sphere can be determined using:\[ KE_{initial} = KE_{closest} + PE_{closest} \]Where \( KE = \frac{1}{2}m v^2 \) and \( PE = qV \). Solve for initial speed \( v \).

Key Concepts

Uniform charge distribution

In electrostatics, a uniform charge distribution means that charge is spread out evenly throughout a given space, like the volume or surface of a sphere. For this exercise, the charge of 5.00 mC is spread uniformly over the volume of the insulating sphere with a radius of 12.0 cm. This uniform charge distribution is essential for calculating the electric field and potential inside the sphere.

When charges are distributed uniformly, such as in our large sphere, it creates a symmetrical electric field. The symmetry implies that calculations involving electric fields become a bit simpler because they rely on consistent patterns. This is vital because it allows us to determine other aspects like the electric potential more easily.

Electric field inside a sphere

Notably, electric fields behave differently within a uniformly charged sphere compared to outside. Inside the sphere, the electric field is not zero but instead varies linearly with distance from the center.

The formula to calculate the electric field inside a sphere is: \[ E = \frac{1}{4\pi\varepsilon_0}\cdot\frac{Qr}{R^3} \] where:

• \( Q \) is the total charge of the sphere.
• \( r \) is the distance from the center.
• \( R \) is the sphere's radius.
• \( \varepsilon_0 \) is the vacuum permittivity.

This equation shows that the electric field is proportional to the distance from the center, increasing linearly up to the surface of the sphere. Utilizing this property allows us to understand how the electric field changes as our small sphere moves inside the larger one, which is necessary for accurately predicting its motion.

Conservation of energy

The conservation of energy is a core physics principle stating that energy cannot be created or destroyed but only transformed from one form to another. In the context of this problem, it refers to the conservation of mechanical energy, including kinetic and potential energy, as the small sphere moves under electrostatic forces.

Initially, when the small sphere is far from the large one, its potential energy due to the electric field is practically zero. When it approaches closer, some of its initial kinetic energy is converted into electric potential energy.

The equation summarizing this is:\[ KE_{\text{initial}} = KE_{\text{closest}} + PE_{\text{closest}} \] This relationship helps us determine the initial speed needed for the small sphere to reach the desired closest distance to the center of the large sphere. It balances the reduction in kinetic energy with the increase in potential energy.

Electric potential

Electric potential, measured in volts, is a scalar quantity that represents the electric potential energy per unit charge at a point in space. Inside uniformly charged spheres, the electric potential is positive and decreases as one moves from the center to the surface.

The expression for electric potential inside a sphere is:\[ V = \frac{1}{4\pi\varepsilon_0}\left( \frac{3Q}{2R} - \frac{Qr^2}{2R^3} \right) \]where

• \( V \) is the potential at a distance \( r \) from the center.
• \( Q, R, \) and \( \varepsilon_0 \) maintain the same meanings as before.

This equation shows how the potential decreases from the center toward the surface. To solve this problem, you calculate the potential at the closest approach distance (\( r = 4 \) cm), which gives the potential energy needed for energy conservation equations. Calculating these values provides insight into how much energy the small sphere needs to penetrate the larger sphere up to the specified limit.