Three charges Q1, Q2, and Q3, each equal to 6.4 × 10^-¹⁹ C, are in a straight line. The distance between neighboring charges is 60 nm. The magnitude of the electric field at P, which is 80 nm from Q2 on a line at right angles to the line between Q1 and Q3, is

Question

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The electric field at point P will be the vector sum of the electric fields at that point due to the individual charges Q1, Q2, and Q3. We can calculate the magnitude of the electric field due to each charge using the formula:

E = kQ/r^2

where k is Coulomb's constant (9 × 10^9 N·m^2/C^2), Q is the charge, and r is the distance from the charge to the point P.

For Q1 and Q3, the distance is:

sqrt((80 nm)^2 + (30 nm)^2) = 86.8 nm

Therefore, the magnitude of the electric field due to Q1 and Q3 at point P is:

E1 = E3 = kQ/r^2 = 9 × 10^9 × (6.4 × 10^-10 C)/(86.8 × 10^-9 m)^2 = 1.9 × 10^5 N/C

For Q2, the distance is simply:

60 nm

Therefore, the magnitude of the electric field due to Q2 at point P is:

E2 = kQ/r^2 = 9 × 10^9 × (6.4 × 10^-10 C)/(60 × 10^-9 m)^2 = 2.2 × 10^5 N/C

Note that the electric field due to Q2 is larger than that due to Q1 or Q3 because point P is closer to Q2.

Now we need to add the electric fields due to each charge vectorially. Since the charges are in a straight line and Q1 and Q3 are equidistant from Q2, the electric fields due to Q1 and Q3 will cancel out in the x-direction (i.e. along the line connecting the charges), leaving only the y-component of those fields.

We can use trigonometry to find the y-component of each electric field:

sin θ = (30 nm)/86.8 nm = 0.345

cos θ = (80 nm)/86.8 nm = 0.921

The y-component of E1 and E3 is:

E1y = E1 sin θ = 1.9 × 10^5 × 0.345 = 6.5 × 10^4 N/C

E3y = E3 sin θ = 1.9 × 10^5 × 0.345 = 6.5 × 10^4 N/C

The x-component of E2 and the y-component of all three electric fields are in the same direction, so we can simply add them up:

Ey = E1y + E2 + E3y = (6.5 × 10^4 + 2.2 × 10^5 + 6.5 × 10^4) N/C = 3.4 × 10^5 N/C

Finally, we can use the Pythagorean theorem to find the magnitude of the total electric field at point P:

E = sqrt((Ex)^2 + (Ey)^2) = sqrt((0)^2 + (3.4 × 10^5 N/C)^2) = 3.4 × 10^5 N/C

So the answer is D, 1.9 × 10^5 N/C.

Three Charges Q1, Q2, and Q3, Each Equal to 6