Two charges +q and −q are separated by a distance d=50 cm. The electric field lines for this system form the the well-known pattern showed in the figure below. Consider two lines L1 and L2 lying on the same plane and leaving the positive charge at α=±45∘. What is the maximum distance D between the lines in centimeters.

use diff eq. with coulombs law E = kq/r^2 r^

Solve dy/dx = E_net,y/E_net,x

thats not a good technique as DE will get too complicated

what are the steps to solve this??

is the answer 25?

To determine the maximum distance D between the lines L1 and L2, we need to analyze the properties of electric field lines for a system of two charges.

The electric field lines originate from the positive charge (+q) and terminate on the negative charge (-q). They always start perpendicular to the surface of the positive charge and end perpendicular to the surface of the negative charge.

Since the field lines for this system form a well-known pattern, we can analyze its symmetry to find the maximum distance between the lines L1 and L2.

Let's break down the problem step by step:

Step 1: Determine the direction of the electric field lines.

The electric field lines between opposite charges always point away from the positive charge (+q) and towards the negative charge (-q). In this case, they will extend radially outward from the positive charge and radially inward towards the negative charge.

Step 2: Analyze the symmetry of the system.

In this case, we have two lines L1 and L2 that leave the positive charge at α=±45∘. Since they are symmetrically located with respect to the positive charge, we can horizontally draw lines starting from the positive charge at ±45∘.

Step 3: Measure the distance D.

The maximum distance D between the lines L1 and L2 can be determined by observing the pattern of the electric field lines. Based on the symmetry of the system, the distance D will be measured between the tangent points of the field lines where they cross the horizontal lines at ±45∘.

Step 4: Calculate the maximum distance D.

Given that the distance between the charges is d = 50 cm, we can visualize that the electric field lines form concentric circles around the positive charge. The distance D can be calculated by considering the angle α=±45∘ and using basic trigonometry.

The tangent of ±45∘ is 1, so we can set up a right triangle. The vertical leg represents D, the horizontal leg represents d/2 (half the distance between the charges), and the hypotenuse represents the radius of the circle (which is not relevant to the problem).

Using the tangent formula, we have:

tan(α) = vertical leg / horizontal leg
tan(45∘) = D / (d/2)

Since tan(45∘) = 1, we can simplify the equation:

1 = D / (d/2)
2 = D / d
D = 2d

Given that d = 50 cm, the maximum distance D between the lines L1 and L2 is:

D = 2d = 2 * 50 cm = 100 cm

Therefore, the maximum distance D between the lines is 100 centimeters (cm).