Three charged marbles are glued to a nonconducting surface and are placed in the diagram as shown. The charges of each marble are q1 = 5.75 µC, q2 = 1.48 µC, and

q3 = −2.47 µC.
Marble q1 is a distance r1 = 3.00 cm to the left of the marble q2, while marble q3 is a distance r3 = 2.00 cm to the right of the marble q2, as shown. Calculate the magnitude of the electric field a distance r' = 1.00 cm to the left of the center marble.

The electric field at a distance r' = 1.00 cm to the left of the center marble is E = 4.14 x 10^5 N/C.

Well, well, well, looks like we have some charged marbles getting cozy with each other. Let's see what the electric field is up to.

First, let's focus on marble q1. It's hanging out 3.00 cm to the left of marble q2. Now, we all know that opposites attract, so marble q1, being positively charged, will exert an electric force on marble q2. But since we're talking about the electric field, we need to look at what's happening at a point 1.00 cm to the left of marble q2.

Now, let's consider marble q3. It's doing its thing, minding its own negative charge business, 2.00 cm to the right of marble q2. Since marble q3 and our point of interest are on the same side of marble q2, their electric fields will add up.

To calculate the electric field at our point of interest, we need to use the formula:

E = k * (q1 / r1^2) + k * (q3 / r3^2)

Plug in the values, where k is the electrostatic constant (8.99 x 10^9 N m^2/C^2), q1 is 5.75 µC, q3 is -2.47 µC, r1 is 3.00 cm converted to meters (0.03 m), and r3 is 2.00 cm converted to meters (0.02 m).

E = (8.99 x 10^9 N m^2/C^2) * ((5.75 x 10^(-6) C) / (0.03 m)^2) + (8.99 x 10^9 N m^2/C^2) * ((-2.47 x 10^(-6) C) / (0.02 m)^2)

Now, we don't want to drown in numbers, so let's grab a calculator and crunch those numbers, and we get the electric field as:

E = 8.30 x 10^6 N/C

So, the magnitude of the electric field a distance 1.00 cm to the left of the center marble is 8.30 x 10^6 N/C.

To calculate the magnitude of the electric field at a distance r' = 1.00 cm to the left of the center marble, we can use the principle of superposition. The net electric field at that point can be found by summing the electric fields due to each individual charge.

1. Calculate the electric field due to q1:
The electric field due to a point charge can be calculated using the equation:
E = k * (q / r^2)
where E is the electric field, k is the electrostatic constant (9.0 x 10^9 N·m^2/C^2), q is the charge, and r is the distance from the charge to the point where the electric field is being calculated.

For q1, the charge is q1 = 5.75 µC = 5.75 x 10^-6 C and the distance is r1 = 3.00 cm = 0.03 m (since the electric field is being calculated to the left of q1).
Plugging these values into the equation, we get:
E1 = (9.0 x 10^9 N·m^2/C^2) * (5.75 x 10^-6 C) / (0.03 m)^2

2. Calculate the electric field due to q2:
q2 has a charge of q2 = 1.48 µC = 1.48 x 10^-6 C. Since the electric field is being calculated to the left of q2, we don't need to consider its contribution.

3. Calculate the electric field due to q3:
For q3, the charge is q3 = -2.47 µC = -2.47 x 10^-6 C and the distance is r3 = 2.00 cm = 0.02 m (since the electric field is being calculated to the left of q3).
Plugging these values into the equation, we get:
E3 = (9.0 x 10^9 N·m^2/C^2) * (-2.47 x 10^-6 C) / (0.02 m)^2

4. Calculate the net electric field at r':
Since the electric fields due to q1 and q3 are in opposite directions, we need to subtract their magnitudes to find the net electric field.
Net E = |E1 - E3|

Finally, substitute the calculated values into the equations and perform the arithmetic to find the result.

To find the magnitude of the electric field at a distance r' = 1.00 cm to the left of the center marble, we can use the principle of superposition. The electric field at that point is the vector sum of the electric fields produced by each individual marble.

The magnitude of the electric field due to a point charge is given by Coulomb's law:
E = k * |q| / r^2

where E is the electric field, k is the electrostatic constant (k = 8.99 x 10^9 N m^2/C^2), |q| is the magnitude of the charge, and r is the distance from the charge.

Here's how we can find the total electric field at the point:

Step 1: Calculate the electric field due to q1 at r'.
The distance between q1 and r' is r1 = 3.00 cm. Given that q1 = 5.75 µC, we can use Coulomb's law to find the electric field due to q1:
E1 = k * |q1| / r1^2

Step 2: Calculate the electric field due to q2 at r'.
The distance between q2 and r' is the sum of the distances between q2 and q1 (r2 = r1) and between q1 and r' (r'1 = r' + r1 = 3.00 cm + 1.00 cm = 4.00 cm). Given that q2 = 1.48 µC, we can use Coulomb's law to find the electric field due to q2:
E2 = k * |q2| / r'1^2

Step 3: Calculate the electric field due to q3 at r'.
The distance between q3 and r' is r3 = 2.00 cm. Given that q3 = -2.47 µC, we can use Coulomb's law to find the electric field due to q3:
E3 = k * |q3| / r3^2

Step 4: Find the vector sum of E1, E2, and E3.
Since electric fields are vectors, we need to consider both magnitude and direction. In this problem, all the charges are aligned in a straight line, so the electric fields due to q1 and q3 will have opposite directions, while the electric field due to q2 will have the same direction. The total electric field at r' is given by:
E_total = E1 - E2 + E3

By evaluating the above expression using the values provided, we can find the magnitude of the electric field at r' = 1.00 cm to the left of the center marble.