Electric fields

Question

Often we have distributions of charge for which integrating to find the electric field may not be possible in practice. In such cases, we may be able to get a good approximate solution by dividing the distribution into small but finite particles and taking the vector sum of the contributions of each. To see how this might work, consider a very thin rod of length L = 22 cm

 with uniform linear charge density 𝜆 = 74.0 nC/m.

 Estimate the magnitude of the electric field at a point P a distance d = 11 cm

 from the end of the rod by dividing it into n segments of equal length as illustrated in the figure below for n = 4.

 Treat each segment as a particle whose distance from point P is measured from its center. Find estimates of E_P for n = 1, 2, 4, and 8

 segments.

Answer 1

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To estimate the magnitude of the electric field at point P due to a thin rod with uniform linear charge density, we can divide the rod into small segments and calculate the contribution of each segment to the electric field at point P. By summing up these contributions, we can obtain an approximate solution.

Let's consider a very thin rod of length L = 22 cm with a uniform linear charge density 𝜆 = 74.0 nC/m. We want to estimate the magnitude of the electric field at point P, which is located at a distance d = 11 cm from one end of the rod.

To begin, we divide the rod into n segments of equal length. For simplicity, let's assume that each segment has a length of L/n. We will treat each segment as a particle and calculate its contribution to the electric field at point P.

The electric field contribution from each segment can be calculated using Coulomb's law:

dE = k * (dq / r^2) * u

where dE is the electric field contribution from a segment, k is Coulomb's constant (8.99 × 10^9 N m^2/C^2), dq is the charge of the segment, r is the distance from the segment to point P, and u is a unit vector pointing from the segment towards point P.

Since we have a uniform linear charge density, we can express dq in terms of 𝜆 and dl, where dl is an infinitesimally small length element of the rod:

dq = 𝜆 * dl

Now, let's calculate the electric field contribution for each segment and sum them up to estimate the magnitude of the electric field at point P for different values of n.

For n = 1:

In this case, we consider the entire rod as one segment. The distance from this segment to point P is half of L. Therefore, the electric field contribution from this segment is:

dE = k * (dq / r^2) * u

= k * (𝜆 * dl / (L/2)^2) * u

= k * (𝜆 * dl / (L/2)^2) * (1, 0, 0)

Integrating over the entire length of the rod, we have:

E = ∫ dE

= ∫ k * (𝜆 * dl / (L/2)^2) * (1, 0, 0)

= k * 𝜆 / (L/2)^2 * ∫ dl

= k * 𝜆 / (L/2)^2 * L

= k * 𝜆 / (L/2)

= k * 𝜆 / d

Substituting the given values, we have:

E = (8.99 × 10^9 N m^2/C^2) * (74.0 × 10^-9 C/m) / (11 × 10^-2 m)

E ≈ 6.49 × 10^4 N/C

For n = 2:

In this case, we divide the rod into two equal segments. The distance from each segment to point P is L/4. Therefore, the electric field contribution from each segment is:

dE = k * (dq / r^2) * u

= k * (𝜆 * dl / (L/4)^2) * u

= k * (𝜆 * dl / (L/4)^2) * (1, 0, 0)

Integrating over half of the length of the rod, we have:

E = ∫ dE

= ∫ k * (𝜆 * dl / (L/4)^2) * (1, 0, 0)

= k * 𝜆 / (L/4)^2 * ∫ dl

= k * 𝜆 / (L/4)^2 * (L/2)

= 2 * k * 𝜆 / L

= 2 * (8.99 × 10^9 N m^2/C^2) * (74.0 × 10^-9 C/m) / (22 × 10^-2 m)

E ≈ 5.17 × 10^4 N/C

For n = 4:

In this case, we divide the rod into four equal segments. The distance from each segment to point P is L/8. Therefore, the electric field contribution from each segment is:

dE = k * (dq / r^2) * u

= k * (𝜆 * dl / (L/8)^2) * u

= k * (𝜆 * dl / (L/8)^2) * (1, 0, 0)

Integrating over a quarter of the length of the rod, we have:

E = ∫ dE

= ∫ k * (𝜆 * dl / (L/8)^2) * (1, 0, 0)

= k * 𝜆 / (L/8)^2 * ∫ dl

= k * 𝜆 / (L/8)^2 * (L/4)

= 4 * k * 𝜆 / L

= 4 * (8.99 × 10^9 N m^2/C^2) * (74.0 × 10^-9 C/m) / (22 × 10^-2 m)

E ≈ 1.03 × 10^5 N/C

For n = 8:

In this case, we divide the rod into eight equal segments. The distance from each segment to point P is L/16. Therefore, the electric field contribution from each segment is:

dE = k * (dq / r^2) * u

= k * (𝜆 * dl / (L/16)^2) * u

= k * (𝜆 * dl / (L/16)^2) * (1, 0, 0)

Integrating over an eighth of the length of the rod, we have:

E = ∫ dE

= ∫ k * (𝜆 * dl / (L/16)^2) * (1, 0, 0)

= k * 𝜆 / (L/16)^2 * ∫ dl

= k * 𝜆 / (L/16)^2 * (L/8)

= 8 * k * 𝜆 / L

= 8 * (8.99 × 10^9 N m^2/C^2) * (74.0 × 10^-9 C/m) / (22 × 10^-2 m)

E ≈ 2.06 × 10^5 N/C

Therefore, the estimates of the magnitude of the electric field at point P for n = 1, 2, 4, and 8 segments are approximately:

For n = 1: E ≈ 6.49 × 10^4 N/C

For n = 2: E ≈ 5.17 × 10^4 N/C

For n = 4: E ≈ 1.03 × 10^5 N/C

For n = 8: E ≈ 2.06 × 10^5 N/C