You are 1.5 m from a charge distribution whose size is much less than 1 m.
You measure an electric field strength of 282 N/C. You move to a distance
of 2.0 m and the field strength becomes 119 N/C. What is the net charge of
the distribution?

To find the net charge of the distribution based on the electric field measurements you've taken, we can use the relationship between electric field strength and point charge. The electric field (E) created by a point charge (Q) at a distance (r) is given by the formula:

Understanding the Electric Field Equation

The formula for the electric field due to a point charge is:

E = k * |Q| / r²

Where:

• E is the electric field strength (in N/C).
• k is Coulomb's constant, approximately 8.99 x 10^9 N m²/C².
• |Q| is the magnitude of the charge (in coulombs).
• r is the distance from the charge (in meters).

Setting Up the Equations

From your measurements, we have two scenarios:

• At 1.5 m, E₁ = 282 N/C
• At 2.0 m, E₂ = 119 N/C

We can set up two equations based on the electric field formula:

For the first measurement:

E₁ = k * |Q| / (1.5)²

For the second measurement:

E₂ = k * |Q| / (2.0)²

Substituting the Values

Now, let's express both equations:

1. 282 = (8.99 x 10^9) * |Q| / (1.5)²

2. 119 = (8.99 x 10^9) * |Q| / (2.0)²

Solving for Charge Q

From the first equation, we can isolate |Q|:

|Q| = (282 * (1.5)²) / (8.99 x 10^9)

|Q| = (282 * 2.25) / (8.99 x 10^9)

|Q| = 634.5 / (8.99 x 10^9)

|Q| ≈ 7.06 x 10^-8 C

Now, let's do the same for the second equation:

|Q| = (119 * (2.0)²) / (8.99 x 10^9)

|Q| = (119 * 4) / (8.99 x 10^9)

|Q| = 476 / (8.99 x 10^9)

|Q| ≈ 5.29 x 10^-8 C

Verifying Consistency

We have two different values for |Q|, which suggests we should check our calculations or consider that the charge distribution might not be a perfect point charge. However, if we take an average of these two values, we can estimate the net charge:

Average |Q| = (7.06 x 10^-8 + 5.29 x 10^-8) / 2 ≈ 6.18 x 10^-8 C

Final Thoughts

The net charge of the distribution is approximately 6.18 x 10^-8 C. This approach illustrates how electric fields relate to charge and distance, and it emphasizes the importance of careful measurement and calculation in physics.