Four equal negative point charges are located at the corners of a square, their positions in the xy-plane being (1,1), (−1,1), (−1,−1), (1,−1).

The electric field on the x-axis at (1,0) points in the same direction as:

A) -j

B) i

C) -i

D) j

c) -i

Explanation:

Since the particles carry the same charges they will all repel each other,

Since the particles are at equal distance from each other, the magnitude of their repelling forces on each other will be equal assuming they are all of equal mass.

Each perticle will move away at a 90° direction from the center and this puts the electric field at the center to sum up to zero.

All field will point towards the point of least potential hence the electric field at point (1, - 1) will point towards -I direction which is along the x-axis towards the zero point.

Mathematically, force on each particle Fp = -Qq/r^2 (coulumb's law)

Assuming each has a charge of 1C

The force field between point (1, 0) and point (-1, 1) or (-1, - 1) will vary as inverse square of their distance,

Since they are 5 units apart (length of their diagonal), they experience a repulsion of the magnitude 0.04 (1/25)

For points (1, 0) and (1, 1) or (1, - 1) the magnitude is 1 since they are 1 unit apart. This is 25 times stronger than for the above.

This means the field will move towards the lesser field which is toward -i

See image below

Result is C,  -i^

Explanation:

The electric field is a vector quantity given by

           E = k ∑

Where k is the Coulomb constant that is worth 8,988 10⁹ N / m² C², q is the charge and r is the distance from the charge to the point of interest

Let's apply this formula to our case

        E = E1 + E2 + E3 + E4

the bold are vectors

The point where we want the field is r₀ = 1 i ^ + 0 j ^. Let's look for the distances

        r² = (x-x₀)² + (y-y₀)²

        r₁² = (1-1)² + (1-0)²

        r₂² = (-1-1)² + (1-0)²

        r₃² = (-1-1)² + (-1-0)²

        r₄² = (1-1)² + (-1-0)²

        r₁² = 1

        r₂² = 5

        r₃² = 5

        r₄² = 1

The charge proof is always considered positive

Let's analyze the direction of each field

²E1 between the charge in position 1 and the point of interest the force is attractive, charge with different sign directed on the y-axis (vertical), whereby E₁ = E₁ j ^

E4 between charge 4 and the test charge, incoming field, directed on the vertical axis and E₄ = -E₄ j ^

E2 between the charge in 3 and the test charge in r, the field is incoming, with the direction on the line between 3 and 1, let's look with trigonometry

           tan θ₂ = y / x

           θ₂ = tan⁻¹ y / x

           θ₂ = tan⁻¹ 1 / 2

           θ₂ = 26.57º

E3 between charge 3 and test carge in ro, incoming field, with address

          tan θ₃ = y / x

          θ₃ = tan⁻¹ y / x

          θ₃ = tan⁻¹ (-1/2)

          θ₃ = -26.57º

From the symmetry and the distances we see that the field E₁ and E₄ are of the same magnitude and opposite direction whereby their sum is zero.

Fields E2 and E3 have components x and y, since the distances are equal the magnitude of the two fields is equal. The sum of the components in y is zero and the components in x

         E_total = 2 E₂ cos 26.57

        Total_ = 2 k q / r₂² cos 26.57 (- i ^)

        Total_ = 2 8,988 10⁹ q / 5 cos 26.57 (- i ^)

        Total_ = 16.08 10⁹ q (-i ^)

The address of this total field is –i ^

Result is C

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