A dipole consists of two charges +q and -q, held apart by a rod of length s, as shown in the diagram. If q = 9 nC and s = 2 mm, what is the magnitude of the electric field due to the dipole at location A, a distance d = 5 cm from the dipole?
E=____N/C
What is the magnitude of the electric field due to the dipole at location B, a distance d = 5 cm from the dipole?
E=____N/C
To calculate the magnitude of the electric field due to a dipole at a certain location, you can use the formula:
E = (1 / 4πε₀) * (( p / r³ ) * (3cos²θ - 1))
Where:
- E is the magnitude of the electric field
- ε₀ is the permittivity of free space, approximately equal to 8.85 x 10⁻¹² C²/N m²
- p is the dipole moment, given by p = q * s
- r is the distance between the location and the dipole
- θ is the angle between the dipole axis and the line connecting the dipole to the location
Let's calculate the magnitude of the electric field at location A:
1. Calculate the dipole moment:
p = q * s
p = (9 nC) * (2 mm)
p = 18 nC mm
2. Convert the dipole moment to SI units (C m):
p = 18 nC mm
p = 18 × 10⁻⁹ C × 0.002 m
p = 3.6 × 10⁻¹² C m
3. Calculate the distance between the location and the dipole:
r = 5 cm
r = 0.05 m
4. Calculate the angle θ between the dipole axis and the line connecting the dipole to the location:
At location A, θ = 0 degrees (since line A is along the axis of the dipole)
5. Plug the values into the formula for the magnitude of the electric field:
E = (1 / (4πε₀)) * ((p/r³) * (3cos²θ - 1))
E = (1 / (4π * 8.85 x 10⁻¹² C²/N m²)) * ((3.6 x 10⁻¹² C m / (0.05 m)³) * (3cos²(0 degrees) - 1))
Now, calculate each part of the formula step by step:
- Calculate (1 / (4π * 8.85 x 10⁻¹² C²/N m²)):
(1 / (4π * 8.85 x 10⁻¹² C²/N m²)) ≈ 9 x 10⁹ N⁻¹ m² C²
- Calculate (3.6 x 10⁻¹² C m / (0.05 m)³):
(3.6 x 10⁻¹² C m / (0.05 m)³) ≈ 1.44 x 10⁻⁷ C m⁴ N⁻¹
- Calculate (3cos²(0 degrees) - 1):
(3cos²(0 degrees) - 1) = 3 - 1 = 2
- Now multiply all the parts together:
E = (9 x 10⁹ N⁻¹ m² C²) * (1.44 x 10⁻⁷ C m⁴ N⁻¹) * 2
E = 2.592 x 10³ N/C
So, the magnitude of the electric field due to the dipole at location A, a distance of 5 cm from the dipole, is approximately 2.592 x 10³ N/C.
Now, let's calculate the magnitude of the electric field at location B:
1. The distance between the location and the dipole (r) is the same as before: 0.05 m.
2. The angle θ between the dipole axis and the line connecting the dipole to the location is also the same: 0 degrees.
3. Plug the values into the formula for the magnitude of the electric field:
E = (1 / (4πε₀)) * ((p/r³) * (3cos²θ - 1))
E = (1 / (4π * 8.85 x 10⁻¹² C²/N m²)) * ((3.6 x 10⁻¹² C m / (0.05 m)³) * (3cos²(0 degrees) - 1))
Following the calculations from before, the result will be the same:
E ≈ 2.592 x 10³ N/C
So, the magnitude of the electric field due to the dipole at location B, a distance of 5 cm from the dipole, is also approximately 2.592 x 10³ N/C.