Why does the unit vector r-hat always point away from a charge?

In summary: The unit vector is irrelevant to the direction from q1 to q2. The unit vector points in the direction of the force that q1 exerts on q2. That's all. I was mistaken before.In summary, in the textbook, it is stated that when given two positive charges of equal magnitude, the unit vector r-hat always points away from the first charge. This is because the unit vector describes the direction of the field from the first charge, which is outward. When the charges have opposite signs, the unit vector can point either towards or away from the first charge, depending on the location of the origin of the coordinate system. The unit vector is not tied to a
  • #1
negation
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In my textbook, it states that given q1 and q2 where both charges are positive and of equal magnitude, the unit vector r-hat always points away from q1. Why is this so?
What if both charges are of opposite charge? Does r-hat still points away?
 
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  • #2
negation said:
In my textbook, it states that given q1 and q2 where both charges are positive and of equal magnitude, the unit vector r-hat always points away from q1. Why is this so?
What if both charges are of opposite charge? Does r-hat still points away?
Presumably the unit vector is describing the field from q1, which is outward. (The field from a negative charge would point inward.)

Do you have a diagram of what they are describing?
 
  • #3
negation said:
In my textbook, it states that given q1 and q2 where both charges are positive and of equal magnitude, the unit vector r-hat always points away from q1. Why is this so?
What if both charges are of opposite charge? Does r-hat still points away?

Er.. it doesn't have to! It depends on where you put the origin of your coordinate axis. If you put the origin right in the middle in between the two charges, then the radial unit vector points AT each of the charge when you are in between the two charges, and then away from the charges if you are beyond the two charges.

The radial unit vector is a function of where you define the origin of your coordinate system. It has nothing to do with being tied to a particular charge. In other words, it is math. However, placing it in an appropriate location can simplify the mathematics. That is why it is usually placed at the location of the source charge!

Zz.
 
  • #4
negation said:
In my textbook, it states that given q1 and q2 where both charges are positive and of equal magnitude, the unit vector r-hat always points away from q1. Why is this so?
What if both charges are of opposite charge? Does r-hat still points away?
The unit vector r-hat is a geometric quantity, not a physical quantity. If one of the charges is at the origin and you are employing spherical coordinates, in spherical coordinates the radial unit vector r-hat points away from the origin.

Chet
 
  • #5
Doc Al said:
Presumably the unit vector is describing the field from q1, which is outward. (The field from a negative charge would point inward.)

Do you have a diagram of what they are describing?

Yes I do. But how do I upload it? I do not have a scanner so the only option is to utilize my phone's camera.

10014440_10201910396416688_7602218934664254168_o.jpg
 
  • #6
Chestermiller said:
The unit vector r-hat is a geometric quantity, not a physical quantity. If one of the charges is at the origin and you are employing spherical coordinates, in spherical coordinates the radial unit vector r-hat points away from the origin.

Chet

Isn't it the case that if we were measuring the electric force at a point at <5,5> then r-hat would be <5,5>?-that is, r-hat is a vector to provide the coordinate of the point at which we are measuring an electric force?
If the point at which we are measuring the electric force is at <0,0>, where would r-hat point to?
 
  • #7
negation said:
Isn't it the case that if we were measuring the electric force at a point at <5,5> then r-hat would be <5,5>?-that is, r-hat is a vector to provide the coordinate of the point at which we are measuring an electric force?

Move your origin to (-10, 0). What is "r-hat" now?

Zz.
 
  • #8
negation said:
Yes I do. But how do I upload it? I do not have a scanner so the only option is to utilize my phone's camera.

View attachment 68552

It's just a notational convenience to ensure that the formula always gives the correct direction.

It's a unit vector that points away from the other charge.
 
  • #9
ZapperZ said:
Move your origin to (-10, 0). What is "r-hat" now?

Zz.

<-5,5>?
 
  • #10
negation said:
<-5,5>?

I think you have a problem with geometry, not with physics.

Zz.
 
  • #11
negation said:
Isn't it the case that if we were measuring the electric force at a point at <5,5> then r-hat would be <5,5>?

It's a unit vector, so it's magnitude has to be 1, so you have to normalize it too by dividing each component by its magnitude. You'd get [itex] \hat{r} = \left\langle \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right\rangle[/itex]. The purpose of a unit vector is to provide the direction of something, so you can multiply it by the magnitude of something else (in this case, the electric force) to get the force vector with the correct magnitude and direction. (You're multiplying the magnitude of the force by the magnitude of the unit vector, which is 1, so the resulting vector has the same magnitude as that of the force, but also in the same direction as the unit vector.)
 
  • #12
ZapperZ said:
I think you have a problem with geometry, not with physics.

Zz.

If the coordinate shifts by -10, am I not suppose to move the x-coordinate by that amount?
 
  • #13
negation said:
<-5,5>?

Check that you know how to take the magnitude of a vector and what a unit vector is.
 
  • #14
craigi said:
Check that you know how to take the magnitude of a vector and what a unit vector is.

The magnitude of a vector is sqrt(x^2 + y^2+z^2) and a unit vector i-hat represents one unit of a component in the x-direction.
 
  • #15
This thread is moved to the mathematics forum from the General Physics forum, because this is no longer physics.

Zz.
 
  • #16
negation said:
The magnitude of a vector is sqrt(x^2 + y^2+z^2) and a unit vector i-hat represents one unit of a component in the x-direction.

The first part is correct. Check the second part again.
 
  • #17
negation said:
Yes I do. But how do I upload it? I do not have a scanner so the only option is to utilize my phone's camera.

View attachment 68552
Ah, now I see what's going on. Chestermiller is right.

When using Coulomb's law in vector form, that unit vector always points outward from q1. Then you can use the signs of q1 and q2 to determine the direction of the force that q1 exerts on q2. When they have the same sign, the force acts in the direction of the unit vector.

The unit vector just describes the direction from q1 to q2.
 
  • #18
craigi said:
The first part is correct. Check the second part again.

Isn't i-hat one unit component of x?

The mathematical definition is e1 or vector r / magnitude r
 
  • #19
Doc Al said:
Ah, now I see what's going on. Chestermiller is right.

When using Coulomb's law in vector form, that unit vector always points outward from q1. Then you can use the signs of q1 and q2 to determine the direction of the force that q1 exerts on q2. When they have the same sign, the force acts in the direction of the unit vector.

The unit vector just describes the direction from q1 to q2.

and if the force were from q2 to q1? the unit vector would point to the left?

In the second diagram however, the electric force is from q2 to q1 yet the unit vector points to the right of q2.
 
  • #20
negation said:
Isn't i-hat one unit component of x?

The mathematical definition is e1 or vector r / magnitude r

You're close, but this isn't quite correct. Where are you learning this from?
 
  • #21
negation said:
and if the force were from q2 to q1? the unit vector would point to the left?
No, the unit vector describes the direction from q1 to q2. The force may be in the same direction or in the opposite direction, depending on the signs of the charges.

In the second diagram however, the electric force is from q2 to q1 yet the unit vector points to the right of q2.
That's right.
 

1. Why does the unit vector r-hat always point away from a charge?

The unit vector r-hat always points away from a charge because it represents the direction of the electric field at a point due to that charge. The direction of the electric field is always from positive charges to negative charges.

2. Is there any exception to the rule that r-hat points away from a charge?

No, the direction of the unit vector r-hat is always away from a charge regardless of the type or location of the charge. This is a fundamental principle in electrostatics known as the "right-hand rule."

3. How does the magnitude of the charge affect the direction of r-hat?

The magnitude of the charge does not affect the direction of r-hat. The direction of r-hat is solely determined by the type (positive or negative) of the charge and its location relative to the point of interest.

4. Can the direction of r-hat change at different points around a charge?

Yes, the direction of r-hat can change at different points around a charge. This is because the electric field is a vector quantity and its direction can vary depending on the location and orientation of the charge.

5. How is the direction of r-hat related to the force experienced by a test charge?

The direction of r-hat is directly related to the force experienced by a test charge. The electric field is defined as the force per unit charge, and thus the direction of the unit vector r-hat determines the direction of the force on a test charge due to a particular charge.

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