University of Wisconsin Green Bay

The direction of electric field is defined as being radially out from a positive charge and radially in towards a negative charge.

Point charges do not experience an electric field due to their own charge. Therefore, the field on the 2 µC charge is the vector sum of three fields—those due to the -2 µC charge, the 3 µC charge, and the -3 µC charge.

Point charges do not experience an electric force due to their own charge. Therefore, the force on the 2 µC charge is the vector sum of three forces—those due to interactions with the -2 µC charge, the 3 µC charge, and the -3 µC charge.

The direction of electric force is determined by the location of the charges and whether it is attractive (interaction between a + and a – charge) or repulsive (interaction of like charges.)

Isn’t this a Newton’s Second Law problem?

No. In Newton’s Second Law, you are asked to relate the forces on an object to the motion of that object. In this case, you aren’t asked about motion (in fact, the charges are fixed in place by non-specified forces) but rather you are just asked for the electric force and field. This is the equivalent of being asked for gravitational force when you are given masses.

I use E_-3 to refer to the electric field due to the -3 µC charge. At the origin, the direction of this field is from the origin towards the negative charge.

I use E_-2 to refer to the electric field due to the -2 µC charge. At the origin, the direction of this field is from the origin towards the negative charge.

I use E₃ to refer to the electric field due to the +3 µC charge. At the origin, the direction of this field is from the origin away from the positive charge.

The dotted line is merely a guide to correctly show the direction of E₃.

tan θ = (opposite side)/(adjacent side) = (0.10)/(0.20) = 0.50
θ = tan^-1 (0.50) [or inv tan (0.50) depending on your calculator] = 27⁰

As you can see in the figure, the x- and y-components of a vector make up the sides of a right triangle. The vector itself forms the hypotenuse (h). The side of the triangle opposite the angle that you use is given by h sinθ and the side that touches the angle you use is given by h cosθ (soh cah toa)

In this case, the x-component is adjacent to the 27⁰angle, and so is given by E₃cos(27⁰). Likewise, the y-component is opposite to the 27⁰angle and is therefore given by E₃sin(27⁰).

You know this angle is 27⁰ because when two lines intersect (in this case, the x-axis and the guideline between the origin and the 3 µC charge,) the angles opposite to each other are the same.

How do you know the direction of electric field?

Electric field is defined as having the same direction as force on a positive charge. In other words, electric field due to a negative charge points in towards the charge, and electric field due to a positive charge points out away from the charge. So to draw the electric field at any point, sketch a guide line between that point and the charge whose field you want to draw. Then draw the field in or out along that guideline.

Why didn’t you include the field of the 2 µC charge?

An object cannot put a force on itself—it doesn’t feel its own field. So the 2 µC charge experiences a force due to (the field of) the -2 µC charge, the 3 µC charge, and the -3 µC charge, but not due to itself.

How do you know that the angle between E₃ and the –y axis is 27^o?

You probably learned this as “opposite angles are congruent” in geometry. When two lines intersect (in this case, the x-axis and the guideline between the origin and the 3 µC charge,) the angles opposite each other are the same.

I use F_2-3 to refer to the electric force between the 2 µC charge and the -3 µC charge. This interaction is attractive and on a line between the two charges.

I use F_2-3 to refer to the electric force between the 2 µC charge and the +3 µC charge. This interaction is repulsive and on a line between the two charges.

I use F_2-2 to refer to the electric force between the + 2 µC charge and the -2 µC charge. This interaction is attractive and on a line between the two charges.

The dotted line is merely a guide to correctly show the direction of F_2-3.

tan θ = (opposite side)/(adjacent side) = (0.10)/(0.20) = 0.50 θ = tan^-1 (0.50) [or inv tan (0.50 depending on your calculator) = 27⁰

As you can see in the figure, the x- and y-components of a vector make up the sides of a right triangle. The vector itself forms the hypotenuse (h). The side of the triangle opposite the angle that you use is given by h sinθ and the side that touches the angle you use is given by h cosθ (soh cah toa)

In this case, the x-component is adjacent to the 27⁰angle, and so is given by F₂₃cos(27⁰). Likewise, the y-component is opposite to the 27⁰angle and is therefore given by F₂₃sin(27⁰).

You know this angle is 27⁰ because when two lines intersect (in this case, the x-axis and the guideline between the origin and the 3 µC charge,) the angles opposite to each other are the same.

How do you know the direction of electric force?

Electric field is defined as having the same direction as force on a positive charge. In other words, electric field due to a negative charge points in towards the charge, and electric field due to a positive charge points out away from the charge. So to draw the electric field at any point, sketch a guide line between that point and the charge whose field you want to draw. Then draw the field in or out along that guideline.

How do you know that the angle between F₂₃ and the –y axis is 27^o?

You probably learned this as “opposite angles are congruent” in geometry. When two lines intersect (in this case, the x-axis and the guideline between the origin and the 3 µC charge,) the angles opposite each other are the same.

k = 9.0 x 10⁹ N∙m²/C²

Some books use 1/(4πε₀) instead of k. In either case, the value is the same.

The -3 µC charge is 0.10 m away from the location at which you want to know the field.

The -2 µC charge is 0.20 m away from the location at which you want to know the field.

The definition of electric field gives the magnitude of the electric field only. Direction comes from your coordinate system and the understanding that electric field points out from negative charges and in to positive charges. Therefore, it is best to use absolute values only in the equation. If you put in the sign of the charge, it is easy to mistake that for the direction of the field.

The +3 µC charge is 0.224 m away from the location at which you want to know the field.

Electric field is out from a positive charge. In this case, out from the 3 µC charge means that at the location of the 2 µC charge the field is down and to the left. In other words, the x- and y- components are both negative.

Electric field acts in to a negative charge. In this case, in to the -3 µC charge means that at the location of the 2 µC charge the field is straight up. In other words, there is no x-component and the y-component is positive.

E² = E_x² + E_y² is the Pythagorean Theorem applied to the electric field vector.

For the angle shown in the figure, the y-component of electric field is opposite the angle and the x-component is adjacent.

What is a µC?

The Coulomb (C) or unit of charge is quite large. Therefore, charge is often given in smaller units. One micro-Coulomb (µC) is 10^-6 C.

Why did you use positive numbers for all charges?

The definition of electric field gives the magnitude of the electric field only. Direction comes from your coordinate system and the understanding that electric field points out from negative charges and in to positive charges. Therefore, it is best to use absolute values only in the equation. If you put in the sign of the charge, it is easy to mistake that for the direction of the field.

How did you find the distance to the 3 µC charge?

The +3 µC charge is 0.224 m away from the location at which you want to know the field.

How did you get the signs on the x- and y-components of E?

Electric field is out from a positive charge. In this case, out from the 3 µC charge means that at the location of the 2 µC charge the field is down and to the left. In other words, the x- and y- components are both negative.

Can’t I do this in one step by multiplying all of the charges together in the equation?

No. For one thing, each charge is at a different distance away from the point of interest. For another, you need to take the vector sum of the fields or forces. There isn’t a correct short cut.

The definition of electric force gives the magnitude of the force only. Direction comes from your coordinate system and the understanding that electric force is attractive between + and – charges and repulsive between same charges. Therefore, it is best to use absolute values only in the equation. If you put in the sign of the charge, it is easy to mistake that for the direction of the force.

The -3 µC charge is 0.10 m away from the location at which you want to know the force.

The -2 µC charge is 0.20 m away from the location at which you want to know the force.

The +3 µC charge is 0.224 m away from the location at which you want to know the force.

Electric force is repulsive between like charges. In this case, “repelled from the 3 µC charge” means that at the 2 µC charge is pushed down and to the left. In other words, the x- and y- components of force are both negative.

Electric force is attractive between positive and negative charges. In this case, the attraction to the -3 µC charge pulls the 2 µC charge the field straight up. In other words, there is no x-component to this force and and the y-component is positive.

Electric field acts in to a negative charge. In this case, in to the -2 µC charge means that at the location of the 2 µC charge the field is straight to the right. In other words, there is no y-component and the x-component is positive.

F² = F_x² + F_y² is the Pythagorean Theorem applied to the electric force vector.

For the angle shown in the figure, the y-component of electric force is opposite the angle and the x-component is adjacent.

What is a µC?

The Coulomb (C) or unit of charge is quite large. Therefore, charge is often given in smaller units. One micro-Coulomb (µC) is 10^-6 C.

Why did you use positive numbers for all charges?

The definition of electric force gives the magnitude of the electric force only. Direction comes from your coordinate system and the understanding that electric field points out from negative charges and in to positive charges. Therefore, it is best to use absolute values only in the equation. If you put in the sign of the charge, it is easy to mistake that for the direction of the force.

How did you find the distance to the 3 µC charge?

The +3 µC charge is 0.224 m away from the location at which you want to know the field.

How did you get the signs on the x- and y-components of F?

Electric field is out from a positive charge. In this case, out from the 3 µC charge means that at the location of the 2 µC charge the field is down and to the left. In other words, the x- and y- components are both negative.

Can’t I do this in one step by multiplying all of the charges together in the equation?

No. For one thing, each charge is at a different distance away from the point of interest. For another, you need to take the vector sum of the fields or forces. There isn’t a correct short cut.

Isn’t this a Newton’s Second Law problem?

No. In Newton’s Second Law, you are asked to relate the forces on an object to the motion of that object. In this case, you aren’t asked about motion (in fact, the charges are fixed in place by non-specified forces) but rather you are just asked for the electric force and field. This is the equivalent of being asked for gravitational force when you are given masses.

How do you know the direction of an electric field?

Electric field is defined as having the same direction as force on a positive charge. In other words, electric field due to a negative charge points in towards the charge, and electric field due to a positive charge points out away from the charge. So to draw the electric field at any point, sketch a guide line between that point and the charge whose field you want to draw. Then draw the field in or out along that guideline.

How do you know the direction of the force?

Like charges are repelled from each other; negative and positive charges are attracted to each other. So just draw a guide line between the charges and sketch the force along that line either towards or away from the charge of interest (in this case, the 2 µC charge.)

What is a µC?

The Coulomb (C) or unit of charge is quite large. Therefore, charge is often given in smaller units. One micro-Coulomb (µC) is 10^-6 C.

Why didn’t you include the field of the 2 µC charge?

An object cannot put a force on itself—it doesn’t feel its own field. So the 2 µC charge experiences a force due to (the field of) the -2 µC charge, the 3 µC charge, and the -3 µC charge, but not due to itself.

How did you know that the angle between E₃ and the –y axis is also 27^o?

You probably learned this as “opposite angles are congruent” in geometry. When two lines intersect (in this case, the x-axis and the guideline between the origin and the 3 µC charge,) the angles opposite each other are the same.

Can’t I do this in one step by multiplying or adding all of the charges together into a single equation?

No. For one thing, each charge is at a different distance away from the point of interest. For another, you need to take the vector sum of the fields or forces. There isn’t a correct short cut.

Don’t I need to know gravity and the other forces on the charges in part b)?

No, you aren’t asked for the total force on the charge, you are just asked for the total electric force on the charge.