Question
Four dipoles, each consisting of a +10 µC charge and a -10 µC charge, are located in the xy-plane with their centers 1.0 mm from the origin, as shown. A sphere passes through the dipoles, as shown in the figure. What is the electric flux through the sphere due to these dipoles? (ε0 = 8.85 × 10-12C2/N m2)
Qu1-pic1-2020
9.0 × 106 N m2/C
0.00 N m2/C
11 × 105 N m2/C
4.5 × 106 N m2/C
Qu1-pic1-2020
9.0 × 106 N m2/C
0.00 N m2/C
11 × 105 N m2/C
4.5 × 106 N m2/C
Answers
GPT-4o mini
To determine the electric flux through the sphere due to the dipoles, we can use Gauss's law, which states that the total electric flux \(\Phi_E\) through a closed surface is related to the enclosed charge \(Q_{enc}\) by the equation:
\[
\Phi_E = \frac{Q_{enc}}{\varepsilon_0}
\]
where \(\varepsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2\).
In this case, let's consider the characteristics of the dipoles:
- Each dipole consists of a +10 µC and a -10 µC charge.
- Since the dipole consists of equal and opposite charges, the total charge of each dipole is zero.
Now, when dealing with multiple dipoles:
- The total charge enclosed by the sphere (assuming it encloses all four dipoles) is:
\[
Q_{enc} = 4 \times (10 \, \mu C - 10 \, \mu C) = 4 \times 0 = 0
\]
As such, the enclosed charge within the sphere is zero. According to Gauss's Law:
\[
\Phi_E = \frac{Q_{enc}}{\varepsilon_0} = \frac{0}{\varepsilon_0} = 0
\]
Thus, the electric flux through the sphere due to these dipoles is:
\[
\Phi_E = 0 \, \text{N} \cdot \text{m}^2/\text{C}
\]
Therefore, the electric flux through the sphere is:
\[
\boxed{0.00 \, \text{N} \cdot \text{m}^2/\text{C}}
\]
\[
\Phi_E = \frac{Q_{enc}}{\varepsilon_0}
\]
where \(\varepsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2\).
In this case, let's consider the characteristics of the dipoles:
- Each dipole consists of a +10 µC and a -10 µC charge.
- Since the dipole consists of equal and opposite charges, the total charge of each dipole is zero.
Now, when dealing with multiple dipoles:
- The total charge enclosed by the sphere (assuming it encloses all four dipoles) is:
\[
Q_{enc} = 4 \times (10 \, \mu C - 10 \, \mu C) = 4 \times 0 = 0
\]
As such, the enclosed charge within the sphere is zero. According to Gauss's Law:
\[
\Phi_E = \frac{Q_{enc}}{\varepsilon_0} = \frac{0}{\varepsilon_0} = 0
\]
Thus, the electric flux through the sphere due to these dipoles is:
\[
\Phi_E = 0 \, \text{N} \cdot \text{m}^2/\text{C}
\]
Therefore, the electric flux through the sphere is:
\[
\boxed{0.00 \, \text{N} \cdot \text{m}^2/\text{C}}
\]