Point charges -q and +5q are fixed at the points (1,0) and (25,0) respectively in the X-Y plane . At which of the following points is the electric field directed towards the origin?

a)(3,4)

b)(28,0)

c)(-5,0)

d)(5,0)

To determine at which point the electric field is directed towards the origin due to the point charges -q and +5q, we need to analyze the contributions of each charge to the electric field at the specified points. The electric field produced by a point charge is given by the formula:

Electric Field Due to a Point Charge

The electric field \( \mathbf{E} \) created by a point charge \( Q \) at a distance \( r \) is expressed as:

\( \mathbf{E} = k \frac{|Q|}{r^2} \hat{r} \)

where \( k \) is Coulomb's constant, \( |Q| \) is the magnitude of the charge, \( r \) is the distance from the charge to the point of interest, and \( \hat{r} \) is the unit vector pointing away from the charge if it is positive, and towards the charge if it is negative.

Charge Positions and Their Effects

In our scenario, we have:

• Charge -q at (1,0)
• Charge +5q at (25,0)

The electric field due to the negative charge -q will point towards the charge, while the electric field due to the positive charge +5q will point away from it.

Evaluating Each Point

Now, let’s evaluate the electric field direction at each of the given points:

1. Point (3,4)

Calculating the distances:

• Distance from -q: \( r_1 = \sqrt{(3-1)^2 + (4-0)^2} = \sqrt{4 + 16} = \sqrt{20} \)
• Distance from +5q: \( r_2 = \sqrt{(3-25)^2 + (4-0)^2} = \sqrt{484 + 16} = \sqrt{500} \)

Both electric fields will be directed away from +5q and towards -q. The net direction will depend on the magnitudes, but since the positive charge is much larger, the field will not point towards the origin.

2. Point (28,0)

Here, the distances are:

• Distance from -q: \( r_1 = 28 - 1 = 27 \)
• Distance from +5q: \( r_2 = 28 - 25 = 3 \)

The electric field from +5q will point away from it, which means it will point rightward, away from the origin. Thus, the electric field does not point towards the origin.

3. Point (-5,0)

Calculating distances:

• Distance from -q: \( r_1 = -5 - 1 = -6 \) (absolute value is 6)
• Distance from +5q: \( r_2 = -5 - 25 = -30 \) (absolute value is 30)

The electric field from -q will point towards it (to the left), while the field from +5q will point away from it (to the right). The net field will point towards the origin, as the leftward field from -q dominates.

4. Point (5,0)

For this point:

• Distance from -q: \( r_1 = 5 - 1 = 4 \)
• Distance from +5q: \( r_2 = 5 - 25 = -20 \) (absolute value is 20)

The electric field from -q will point towards it (to the left), while the field from +5q will point away from it (to the right). The net field will not point towards the origin.

Final Analysis

After evaluating all points, the only location where the electric field is directed towards the origin is at:

Point (-5,0)