Equipotential surface / electric scalar potential problem (why )

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  • #1
FOIWATER
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equipotential surface / electric scalar potential problem (why!!!)

Homework Statement


A potential field is given by V = 3x^2*y - y*z. Is the following statement valid?

"A unit normal to the equipotential surface V = -8 at P(2,-1,4) is <-0.83,0.55,0.07>"


Homework Equations


Gradient of a scalar field?
dot product?

The Attempt at a Solution


First I filled in P(2,-1,4) to V to make sure P is on the equipotential surface.

(this point P is at the unit vector in question)

So I found another random point which also lies on the equipotential surface, I chose <0,1,8>

I then proceeded to apply dot product to both the unit vector in question, and the new vector I found, but did not get 0 (indicating they are not perp)

But I now believe the equipotential surface need not (is likely not, rather) flat in 3 space? is this assumption correct?

So I proceeded to take the gradient of V, and evaluate that at P. I received <-12,8,1>

I know, due to the definition of the gradient, that the direction of this vector indicates the direction of maximum increase of the scalar field V, which is normal to the equipotential surface at P? (would that make sense?)

I then proceeded to take the dot product of the unit vector in question, and the gradient evaluated at P but did not get 1 as I was expecting, rather, I got 14.43

I am assuming my mistake is assuming the gradient of V evaluated at P yields a vector which is perpindicular to V.

Can some one pls help

Tks

EDIT: I realized after I posted I should not of expected to get 1 with the dot product, but I should expect the product of the length of the two vectors. Also I do not get this
 

Answers and Replies

  • #2
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Homework Statement


A potential field is given by V = 3x^2*y - y*z. Is the following statement valid?

"A unit normal to the equipotential surface V = -8 at P(2,-1,4) is <-0.83,0.55,0.07>"


Homework Equations


Gradient of a scalar field?
dot product?

The Attempt at a Solution


First I filled in P(2,-1,4) to V to make sure P is on the equipotential surface.

(this point P is at the unit vector in question)

So I found another random point which also lies on the equipotential surface, I chose <0,1,8>

I then proceeded to apply dot product to both the unit vector in question, and the new vector I found, but did not get 0 (indicating they are not perp)

But I now believe the equipotential surface need not (is likely not, rather) flat in 3 space? is this assumption correct?

So I proceeded to take the gradient of V, and evaluate that at P. I received <-12,8,1>

I know, due to the definition of the gradient, that the direction of this vector indicates the direction of maximum increase of the scalar field V, which is normal to the equipotential surface at P? (would that make sense?)

I then proceeded to take the dot product of the unit vector in question, and the gradient evaluated at P but did not get 1 as I was expecting, rather, I got 14.43

I am assuming my mistake is assuming the gradient of V evaluated at P yields a vector which is perpindicular to V.

Can some one pls help

Tks

EDIT: I realized after I posted I should not of expected to get 1 with the dot product, but I should expect the product of the length of the two vectors. Also I do not get this
You were perfectly correct in assuming that the gradient of V evaluated at P yields a vector which is perpendicular to the surface of constant V. But, it isn't a unit vector. However, it is pointing in the correct direction. To get the unit normal, you need to divide the gradient of V by its own magnitude.

Chet
 
  • #3
FOIWATER
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Thanks for the prompt reply,
I have a the gradient of V @ point P evaluated as <-12,8,1> and a unit vector in this direction as <-0.0574,0.03827,0.0042846>
Since this is not the unit normal given in the problem statement, I can assume this statement is false? (The textbook says it is a true statement)
 
  • #4
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Thanks for the prompt reply,
I have a the gradient of V @ point P evaluated as <-12,8,1> and a unit vector in this direction as <-0.0574,0.03827,0.0042846>
Since this is not the unit normal given in the problem statement, I can assume this statement is false? (The textbook says it is a true statement)
You have made a calculation error. Recheck your calculations.
 
  • #5
FOIWATER
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V=3x^2*y-yz
gradientV = <6xy,3x^2-z,-y>
@P(2,-1,4) gradientV = <-12,8,1>
length of gradient vector = 209
and unit vector is as above...
 
  • #6
FOIWATER
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aaaaaand i forgot to square root the sum of the squares
 
  • #7
FOIWATER
Gold Member
434
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oh, I got 14.43~!
which is what I got for the dot product of the gradient, and the unit vector given, earlier.
But I guess this makes sense, and I shouldn't be surprised, since knowing the unit vector in question WAS a unit vector, the dot product of the gradient and the unit vector would be the gradient length.

well, thanks, guys...
 

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