Consider this configuration: two charges, +3 C and -1 C, separated by 4 cm. Is there a point along the line joining them where the net electric field is zero? If so, where?

The best way to approach this is to ask the question in three different regions: to the left of both charges (Region I), in between both charges (Region II), and to the right of both charges (Region III). In each region, we will look at the electric field produced by each charge individually and then ask if they cancel out.

Before we start, let's recall the form of the magnitude of the electric field from a point charge:

E = kq/r^{2}

The electric field varies linearly with the magnitude of the charge and is inversely proportional to the square of the distance between the charge and the point where you want to know the field.

In Region I, to the left of both charges, we see the electric field from the -1 C charge shown in blue and the electric field from the +3 C charge shown in red. Even though they point in opposite directions (what we want in order for them to cancel), the red vector is considerably larger than the blue vector. This is because we are **closer** to the **larger** charge. Both the smaller r and the larger q act to make the field from the red charge significantly larger than that from the blue charge.

In Region II, in between the two charges, we see that both vectors point in the same direction so there is no possibility of cancelling out. Can you find a place where they have the same magnitude?

In Region III, we see that they are pointing in opposite directions **and** there is a place in this region where their magnitudes are the same. It is at this point where the net electric field is zero.

What happens at this point? Because **F** = q**E**, if there is no electric field at a point then a test charge placed at that point would feel no force. Try to find that point for these two charges by using this test charge. Make sure that the reasoning above is correct and the force is not zero in Region I or II!

How can we calculate this point exactly? Let's say that the point where the electric field is zero is a distance x away from the +3 C charge. The point is a distance x-4 away from the -1 C charge. If we define right as positive, we can write this as:

k (3 C) / x^{2} - k (1 C) / (x - 4)^{2} = 0

where the minus sign is not the one associated with the charge but the one associated with the **direction** of the field from the charge.

We can cancel out both the k's and the factor of C as well as bring the second term over to the right hand side, we get:

3 / x^{2} = 1 / (x - 4)^{2}

Now cross multiplying and expanding the brackets, we have:

3(x^{2} - 8x + 16) = x^{2} x^{2} - 12x + 24 = 0

When we solve this using the quadratic equation we get two answers: x = 9.46 cm and x = 2.54 cm. The answer that we want is x = 9.46 cm because this represents a point in Region III. The other answer represents the point between the charges where the magnitudes were the same (but remember they didn't cancel because both vectors were pointing in the same direction).

Consider the configuration in Milestone 1: two charges, +2 C and +4 C, separated by a distance of 4 cm. Following the same method as above, where is the electric field zero along the line joining the two points? Use x as the distance from the +2 C charge.

Answers to question:

- x = 1.66 cm

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