Problem 6 - University Physics with Modern Physics

Question

At which of the following locations is the electric field the strongest? a) a point \(1 \mathrm{~m}\) from a \(1 \mathrm{C}\) point charge b) a point \(1 \mathrm{~m}\) (perpendicular distance) from the center of a \(1-\mathrm{m}\) -long wire with \(1 \mathrm{C}\) of charge distributed on it c) a point \(1 \mathrm{~m}\) (perpendicular distance) from the center of a \(1-\mathrm{m}^{2}\) sheet of charge with \(1 \mathrm{C}\) of charge distributed on it d) a point \(1 \mathrm{~m}\) from the surface of a charged spherical shell of charge \(1 \mathrm{C}\) with a radius of \(1 \mathrm{~m}\) e) a point \(1 \mathrm{~m}\) from the surface of a charged spherical shell of charge \(1 \mathrm{C}\) with a radius of \(0.5 \mathrm{~m}\)

Step-by-Step Solution

Answer

(a) 1C point charge, (b) 1C in a 1m long wire, (c) 1C in a 1m² sheet, (d) 1C in a spherical shell with a 1m radius, (e) 1C in a spherical shell with a 0.5m radius. Answer: (a) 1C point charge

Step 1: Calculate electric field for point charge

For option "a", we have a point charge of \(1 \mathrm{C}\) and a distance of \(1 \mathrm{~m}\). Use the formula for a point charge electric field: \(E_a=k\frac{Q}{r^{2}}\) Plug in the values: \(E_a=k\frac{1}{(1)^{2}}\) \(E_a=k\) (where \(k = 8.99 \cdot 10^{9} Nm^{2}C^{-2}\))

Step 2: Calculate electric field for long straight wire

For option "b", we have a \(1-\mathrm{m}\)-long wire with a total charge of 1C and a distance of \(1 \mathrm{~m}\). We first calculate the linear charge density: \(\lambda=\frac{Q}{L}=\frac{1}{1}=1\mathrm{C/m}\) Now, calculate the electric field using the formula for a long straight wire: \(E_b=\frac{1}{2\pi\epsilon_{0}}\frac{\lambda}{d}\) Plug in the values: \(E_b=\frac{1}{2\pi\epsilon_{0}}\frac{1}{1}\)

Step 3: Calculate electric field for infinite sheet

For option "c", we have a \(1-\mathrm{m}^{2}\) sheet with a total charge of 1C and a distance of \(1 \mathrm{~m}\). We first calculate the surface charge density: \(\sigma=\frac{Q}{A}=\frac{1}{1}=1\mathrm{C/m^{2}}\) Now, calculate the electric field using the formula for an infinite sheet: \(E_c=\frac{\sigma}{2\epsilon_{0}}\) Plug in the values: \(E_c=\frac{1}{2\epsilon_{0}}\)

Step 4: Calculate electric field for charged spherical shells

For options "d" and "e", we have charged spherical shells. They both have the same charge of 1C, but different radii. The distances are measured from the surface of the spheres. Since the points of interest are outside the shells, we can use the point charge formula. For option d: \(E_d=k\frac{Q}{r^{2}}\) with \(r=(1+1)\) \(E_d=k\frac{1}{(2)^{2}}\) For option e: \(E_e=k\frac{Q}{r^{2}}\) with \(r=(0.5+1)\) \(E_e=k\frac{1}{(1.5)^{2}}\) Now, compare the electric field magnitudes: \(E_a=k>\frac{1}{2\pi\epsilon_{0}}=E_b>E_c=\frac{1}{2\epsilon_{0}}>E_d=k\frac{1}{(2)^{2}}>E_e=k\frac{1}{(1.5)^{2}}\) The electric field is strongest at a point \(1 \mathrm{~m}\) from a \(1 \mathrm{C}\) point charge (option a).

Key Concepts

Point Charge

When we talk about a point charge, we refer to an idealized model of a charged particle where the entire charge is concentrated at a single point in space. The concept is an essential foundation in understanding electrostatics. Using Coulomb's law, we can easily deduce the electric field produced by a point charge. The formula for the electric field, \(E=k\frac{Q}{r^2}\), reflects the inverse square relationship between the field strength and the distance from the charge, designated as \(r\). This means that as one moves away from a point charge, the electric field strength decreases with the square of the distance.

Due to this property, the electric field strength is strongest at the closest point to the charge, which is why in the given problem, option a, being just 1 meter away, yields the strongest electric field.

Linear Charge Density

Linear charge density, denoted by \(\lambda\), is a measure of how much electric charge is distributed along a line. It's calculated by dividing the total charge \(Q\) by the length \(L\) of the line over which the charge is spread: \(\lambda = \frac{Q}{L}\). In the context of electric fields, when dealing with an infinitely long straight wire, the formula for the electric field becomes \(E = \frac{1}{2\pi\epsilon_0}\frac{\lambda}{d}\), where \(d\) is the perpendicular distance from the wire.

This means the field strength diminishes as one moves away from the wire and also varies with the amount of charge per unit length. For a wire with 1 meter length sporting a total of 1 Coulomb of charge, as in the problem's option b, the electric field can be calculated directly using the provided linear density.

Surface Charge Density

Surface charge density, represented by \(\sigma\), quantifies the amount of charge distributed over a particular area and is given by the formula \(\sigma = \frac{Q}{A}\), where \(A\) is the area over which the charge is spread. This concept is crucial when calculating the electric fields produced by charged planes or sheets.

For an infinite charged sheet, the electric field is uniform and does not decrease with distance, which is different from point charges or lines of charge. The magnitude of the electric field due to a surface charge density can be calculated using \(E = \frac{\sigma}{2\epsilon_0}\). As the problem illustrates for option c, even if the charge is distributed over a 1-square meter area, the field strength at a given distance is determined by the surface charge density.

Charged Spherical Shell

A charged spherical shell is a three-dimensional object with charge distributed uniformly over its surface. An important property of charged spherical shells is that the electric field inside a uniformly charged shell is zero, while outside, it behaves as though the entire charge were concentrated at the center. This allows us to use the point charge formula for calculating the electric field outside the shell.

As in options d and e in the problem, even though the charge quantity remains 1 Coulomb for each shell, the radius plays a critical role in the field's strength at a given distance from the shell's surface. Consequently, a smaller spherical shell with the same charge will exert a stronger electric field at the same distance from its surface than a larger one, which explains the results obtained from the calculations.

Coulomb's Law

Coulomb's law is the cornerstone principle that deals with the electrostatic interaction between electrically charged particles. It provides the quantitative means to calculate the electric force between two point charges as well as the electric field emanating from a single charge. The law states that the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them: \(F = k\frac{|q_1q_2|}{r^2}\), where \(k\) is Coulomb's constant.

Coulomb's law also forms the basis for the calculation of electric field strength; it shows that electric field strength is a vector quantity with both magnitude and direction, where the field lines point away from positive charges and towards negative charges. Understanding this law is essential for solving problems related to electric fields, as it was applied to each option in the problem to determine where the electric field is the strongest.