Potential Difference and Potential Near a Charged Sheet

In summary, the conversation discusses the potential difference between two points, A and B, near and on the same side of a charged sheet with surface charge density +\sigma. The electric field \vec{E} due to the charged sheet has a magnitude of E=\frac{\sigma}{2\epsilon_0} everywhere, and points away from the sheet. Part A asks for the potential difference V_{AB} between points A and B, and Part B asks for the value of the potential at a point V_A at a positive distance y_1 from the surface of the sheet, with the option choices being infinity, negative infinity, 0, or -E*y_1. The solution for Part A is V_{AB}
  • #1
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Homework Statement


Let [tex]{\rm A} = \left(x_1,y_1 \right)[/tex] and [tex]{\rm B} = \left( x_2,y_2 \right)[/tex] be two points near and on the same side of a charged sheet with surface charge density [tex]+ \sigma[/tex] . The electric field [tex]\vec{E}[/tex] due to such a charged sheet has magnitude [tex]E = \frac {\sigma}{2 \epsilon_0}[/tex] everywhere, and the field points away from the sheet, as shown in the diagram. View Figure
184996.jpg


Part A
What is the potential difference [tex]V_{\rm AB} = V_{\rm A} - V_{\rm B}[/tex] between points A and B?

Part B
If the potential at [tex]y = \pm \infty[/tex] is taken to be zero, what is the value of the potential at a point [tex]V_A[/tex] at some positive distance [tex]y_1[/tex] from the surface of the sheet?
choices are:
1. infinity
2. negative infinity
3. 0
4. -E * y_1


Homework Equations


[tex]\int_{\rm B}^{\rm A} \vec{C} \cdot d\vec{\ell} = \int_{x_2}^{x_1} C_x\,dx + \int_{y_2}^{y_1} C_y\,dy
= C_x (x_1 - x_2) + C_y(y_1 - y_2)[/tex]

[tex]V_{\rm AB}= -\int _B^A \vec{E}\cdot d\vec{l}[/tex]



The Attempt at a Solution


Part A.
[tex] V_{\rm AB} = V_{\rm A} - V_{\rm B}= \left(-E\right)\left(y_{1}-y_{2}\right) [/tex]

Part B.
I figure I'd use the equation I got in part A and set the bottom of the E field at y=0.

In which case
V = -E (y_1 - infinity) = infinity

am i on the right track?
 
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  • #2
Looks right to me.
 
  • #3



Yes, you are on the right track. The potential difference between two points is defined as the work done per unit charge in moving a test charge from one point to the other. In this case, the potential difference between points A and B is simply the product of the electric field and the distance between the two points. This is because the electric field is constant everywhere, so the integral simplifies to a simple multiplication.

As for part B, your approach is correct. Since the potential at infinity is taken to be zero, the potential at any point y_1 will be negative infinity, as the electric field decreases with distance from the sheet. Alternatively, you can also use the equation V = -E * y_1 to calculate the potential at point A, where y_1 is the distance from the sheet. This will give the same result of negative infinity.
 

1. What is potential difference?

Potential difference, also known as voltage, is the difference in electric potential energy between two points in an electric field. It is measured in volts (V) and is a measure of the force that drives electric charge to flow from one point to another.

2. How is potential difference related to charged sheets?

Potential difference near a charged sheet is directly proportional to the magnitude of the charge on the sheet and inversely proportional to the distance from the sheet. This means that as the charge on the sheet increases, the potential difference near the sheet also increases, and as the distance from the sheet increases, the potential difference decreases.

3. What is the difference between potential difference and electric potential near a charged sheet?

Potential difference is a measure of the difference in electric potential energy between two points, while electric potential is a measure of the electric potential energy at a specific point in an electric field. In the case of a charged sheet, the electric potential near the sheet is the same at all points, while the potential difference between two points near the sheet may vary.

4. How does the direction of the electric field affect potential difference near a charged sheet?

The direction of the electric field affects the potential difference near a charged sheet. If the electric field is directed towards the sheet, the potential difference will decrease as the distance from the sheet increases. On the other hand, if the electric field is directed away from the sheet, the potential difference will increase as the distance from the sheet increases.

5. Can potential difference and electric potential be negative?

Yes, both potential difference and electric potential can be negative. This indicates that the electric potential energy is decreasing in the direction of the electric field. However, the magnitude of the potential difference and electric potential is usually more important than the sign, as it determines the strength and direction of the electric field.

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