To calculate the electric field at a point located a distance $r_1$ above the center of a square insulating sheet with uniformly distributed charge, we can use the approximation that $r_1 \ll L$, where $L$ is the side length of the square.

Problem Breakdown:

• The sheet has a uniform charge distribution of $Q$ spread over an area of side length $L$.
• We are interested in the electric field at a point located a distance $r_1$ above the center of the sheet.
• The condition $r_1 \ll L$ suggests that the point is very close to the sheet compared to its size, which allows us to treat the system as if it were an infinite plane of charge.

Step 1: Charge Distribution

The surface charge density $\sigma$ (charge per unit area) of the sheet is given by: $\sigma = \frac{Q}{L^2}$ where $L^2$ is the area of the square sheet.

Step 2: Electric Field of an Infinite Sheet of Charge

For an infinite sheet of charge with uniform surface charge density $\sigma$, the electric field at a point near the sheet is given by: $E = \frac{\sigma}{2\epsilon_0}$ where:

• $\epsilon_0$ is the permittivity of free space.
• $\sigma$ is the surface charge density.

This electric field points perpendicular to the surface of the sheet and has the same magnitude on both sides of the sheet. The factor of $\frac{1}{2}$ comes from the fact that the electric field from an infinite sheet is the result of contributions from all points on the sheet.

Step 3: Applying the Approximation

Since $r_1 \ll L$, we can approximate the square sheet as an infinite sheet of charge. Thus, the electric field at a point a distance $r_1$ above the center of the sheet is approximately: $E \approx \frac{\sigma}{2\epsilon_0}$

Step 4: Final Expression for the Electric Field

Substitute the expression for $\sigma$ into the equation: $E \approx \frac{Q}{2\epsilon_0 L^2}$

Thus, the electric field at a distance $r_1$ above the center of the sheet is: $E \approx \frac{Q}{2\epsilon_0 L^2}$

Conclusion:

This is the magnitude of the electric field for the given conditions $r_1 \ll L$, and it is directed perpendicular to the surface of the sheet (either upward or downward depending on the charge distribution).

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Do you need more details about this calculation or have any other questions? Here are some related topics you might find useful:

1. How does the electric field vary with distance for different charge distributions (point charge, dipole, infinite plane)?
2. Derivation of the electric field for an infinite plane of charge.
3. Electric potential due to a uniformly charged square sheet.
4. Electric field due to a circular sheet of charge.
5. How does the electric field change if the observation point is farther from the sheet (when $r_1$ is comparable to $L$)?

Tip: The electric field near a uniformly charged sheet can often be approximated as that of an infinite sheet, simplifying the problem greatly.