Video: Deducing the Charge Distribution of an Arrangement of Charges

Two charges and point 𝑃 form an equilateral triangle as shown in the figure. The two charges are negative, and their magnitude is the same. What is the direction of the electric field at point 𝑃? [A] βˆ’π‘₯ [B] +π‘₯ [C] +𝑦 [D] βˆ’π‘¦ [E] +𝑧

03:28

Video Transcript

Two charges and point 𝑃 form an equilateral triangle as shown in the figure. The two charges are negative, and their magnitude is the same. What is the direction of the electric field at point 𝑃? a) Negative π‘₯, b) positive π‘₯, c) positive 𝑦, d) negative 𝑦, or e) positive 𝑧.

The problem tells us that the two charges and point 𝑃 form an equilateral triangle. This means that the length of each side of the triangle is the same. Let’s call this distance π‘Ÿ. We can say that the distance each negative charge is from point 𝑃 is π‘Ÿ.

The problem also says that the two charges have the same magnitude. We can use the letter 𝑄 to represent the magnitude of both charges. The figure on the right shows us the convention the problem is using for direction. The positive π‘₯-direction is to the right, the positive 𝑦-direction is to the top of the screen, and the positive 𝑧-direction is coming out of the page, as represented by the dot.

We’re trying to determine the direction of the electric field at point 𝑃. To determine the net electric field at point 𝑃, we must first find the net direction of the electric field for each charge individually. Recall that the electric field around a positive point charge is directed radially outward. And that the electric field around a negative point charge is directed radially inward.

We can go back to the figure and draw on the field lines at point 𝑃 due to each negative charge. The blue arrow in the figure shows the direction of the electric field at point 𝑃 due to the negative charge in the bottom-left corner of the triangle. Because the charge is negative, the field will be pointing towards the charge.

Using the figure on the right to describe direction, we can say that the field has components in the negative π‘₯- and negative 𝑦-directions. The green arrow on the figure shows the direction of the electric field at point 𝑃 due to the negative charge in the bottom-right corner of the triangle. Because the charge is negative, the field will once again be pointing towards the charge. This field has components in the positive π‘₯- and negative 𝑦-directions.

Recall that the equation to find the magnitude of the electric field around a point charge is 𝐸 equals π‘˜π‘„ over π‘Ÿ squared. Where 𝐸 is the magnitude of the electric field. π‘˜ is Coulomb’s constant. 𝑄 is the magnitude of the point charge. And π‘Ÿ is the distance from the point charge.

In our problem, we’re told that the magnitude of each charge is the same. And the distance from each charge to point 𝑃 was the same. Because π‘˜ is a constant, and from our problem, we know that the charges are the same. And the distance from each charge to point 𝑃 is the same. We can say that the magnitude of the electric field from each charge at point 𝑃 is also the same.

Looking back at the figure, if the magnitude of the two electric fields at point 𝑃 due to each charge are the same. Then the component of the blue arrow pointing in the negative π‘₯-direction will cancel out with the component of the green arrow pointing in the positive π‘₯-direction. This leaves only the component of each field pointing in the negative 𝑦-direction.

Since both charges have a component of their electric field in the same direction, those components will add together to make a stronger field in that direction. This means that the direction of the electric field at point 𝑃 due to both charges is answer choice d, the negative 𝑦-direction.

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