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Magazine Chapter 23: Problem 78
A charge of \(6.00 \mathrm{pC}\) is spread uniformly throughout the volume of a sphere of radius \(r=4.00 \mathrm{~cm}\). What is the magnitude of the electric field at a radial distance of (a) \(6.00 \mathrm{~cm}\) and (b) \(3.00 \mathrm{~cm}\) ?
Short Answer
Expert verified
At 6.00 cm, the electric field is 1499 N/C; at 3.00 cm, it is 1011 N/C.
Step by step solution
01
Understand the Problem
There is a uniformly charged sphere with a total charge of \(6.00 \mathrm{pC}\) and radius \(4.00 \mathrm{cm}\). We need to find the electric field magnitude at two points: (a) outside the sphere at \(6.00 \mathrm{cm}\) and (b) inside the sphere at \(3.00 \mathrm{cm}\). We will apply Gauss's law to solve this.
02
Calculate Electric Field outside the Sphere
According to Gauss's law, the electric field at a distance \(r\) from a point charge is \(E = \frac{kQ}{r^2}\), where \(k = 8.99 \times 10^9 \mathrm{~N~m^2/C^2}\) is the Coulomb's constant, \(Q\) is the charge, and \(r\) is the radial distance. At \(r = 6.00 \mathrm{~cm}\), we use the entire charge in the sphere:\[E_{6~cm} = \frac{8.99 \times 10^9 \mathrm{~N~m^2/C^2} \times 6.00 \times 10^{-12} \mathrm{C}}{(0.06 \mathrm{~m})^2} = 1499 \mathrm{~N/C}\]
03
Use Gauss's Law Inside the Sphere
Inside a uniformly charged sphere, the electric field is given by the formula \(E = \frac{kQr}{R^3}\), where \(Q\) is the total charge, \(r\) is the radial distance from the center within the sphere, and \(R\) is the sphere鈥檚 radius. At \(r = 3.00 \mathrm{~cm}\),\[E_{3~cm} = \frac{8.99 \times 10^9 \mathrm{~N~m^2/C^2} \times 6.00 \times 10^{-12} \mathrm{C} \times 0.03 \mathrm{~m}}{(0.04 \mathrm{~m})^3} = 1011 \mathrm{~N/C}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gauss's Law
Gauss's Law is a powerful tool in electrostatics that relates electric flux to charge. It states that the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space. This law is expressed mathematically as \[\Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0}\]where \(\Phi_E\) is the electric flux, \(\mathbf{E}\) is the electric field vector, \(d\mathbf{A}\) is a differential area on the closed surface, \(Q_{enc}\) is the enclosed charge, and \(\varepsilon_0\) is the permittivity of free space.
- This law simplifies the calculation of electric fields, especially for symmetric charge distributions.
- It becomes extremely useful for objects like spheres, cylinders, and planes where the symmetrical setup makes the flux calculation straightforward.
Coulomb's Law
Coulomb's Law describes the interaction between two point charges. It tells us that the electric force \(F\) between two charges \(Q_1\) and \(Q_2\) is directly proportional to the product of the charges and inversely proportional to the square of the distance \(r\) between them:\[F = k \frac{|Q_1Q_2|}{r^2}\]where \(k\) is Coulomb's constant, valued at approximately \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\).
- This law forms the foundation of electrostatics, similar to how Newton's law of gravitation governs gravitational interactions.
- It helps explain how charge distributions such as a sphere impact the surrounding space, impacting the electric field.
Uniform Charge Distribution
Uniform charge distribution indicates that charge is spread evenly over a certain volume, surface, or length. For a sphere, this means each small segment of the sphere鈥檚 volume contains the same amount of charge per cubic centimeter.
- In a uniform distribution, the charge density \(\rho\) is constant across the region of the sphere.
- It is calculated as \(\rho = \frac{Q}{V}\), where \(Q\) is the total charge and \(V\) is the total volume of the sphere.
Sphere
A sphere, in the context of electric fields and charges, is often considered due to its symmetry, which simplifies many calculations. The symmetry means any radial line from the center to the surface is identical in terms of electrical properties.
- The radius defines the size of the sphere and influences the volume \(V\) given by \(V = \frac{4}{3}\pi R^3\).
- The spherical symmetry allows assumption of consistent charge density for uniform charge distribution.
Radial Distance
Radial distance refers to the straight-line distance from the center of a sphere to any point either inside or outside the sphere. It plays a crucial role in calculating electric fields using both Coulomb's and Gauss's Laws.
- Outside a uniformly charged sphere, the radial distance \(r\) is considered from the center to the point where the field is measured.
- For points inside, the radial distance becomes crucial in defining part of the volume that influences the electric field, as only the charge within that specific radial distance affects the result.
Electric Field Inside and Outside Sphere
The electric field behaves differently inside and outside a sphere with uniform charge distribution.
- Outside the sphere, at radial distance \(r > R\), the electric field is as if all the charge were concentrated at the center, governed by \(E = \frac{kQ}{r^2}\).
- Inside the sphere at radial distance \(r < R\), the field increases linearly with distance from the center, described by \(E = \frac{kQr}{R^3}\).
