Expression for the magnitude of an electric field

In summary, the problem involves calculating the electric field at point A due to two rings with different positive charges. The magnitude of the electric field is calculated using the line integral formula and the contributions from each ring are added to get the total electric field. The x-component of the electric field is enough to be calculated due to symmetry reasons, and the y and z components are found to be zero at point A. The integral for the upper half of the ring is calculated using the formula $$f(z)=\sqrt{R^2-z^2}$$ and for the lower half using $$\vec{r'}=-\sqrt{R^2-z^2}\hat y+z\hat z$$. Finally, the x-component is multiplied
  • #1
mousey
9
2
Homework Statement
Find an expression for the magnitude of the electric field at point A midway between the two rings of radius R shown in the figure below. The ring on the left has a uniform charge q1 and the ring on the right has a uniform charge q2. The rings are separated by distance d. Assume the positive x axis points to the right, through the center of the rings. (Use any variable or symbol stated above along with the following as necessary: k for Coulomb's constant. Assume q1 is greater than q2, and that both charges are positive.)
(picture in replies)
Relevant Equations
E= (kq)/d^2
k(q_1-q_2)/(R^2+x^2)^3/2
 
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You basically, have to consider one ring, let's say the left one, and calculate the magnitude of the electric field at A due to it, that is calculate the line integral $$\vec{E_1}=\int_{ring 1}\frac{k\rho_1 (\vec{r}-\vec{r'})dr'}{|\vec{r}-\vec{r'}|^3}$$, where $$\vec{r}=\frac{d}{2}\hat x$$,$$\rho_1=\frac{q_1}{2\pi R}$$, $$dr'=\sqrt{1+(f'(z))^2}dz$$ $$f(z)=\sqrt{R^2-z^2}$$, $$\vec{r'}=\sqrt{R^2-z^2}\hat y+z\hat z$$.

It is enough to calculate only the x-component of ##\vec{E_1}##, because the point A lies on the axis that passes through the center of the ring, hence due to symmetry reasons, the y and z components of ##\vec{E_1}## will be zero at point A.

Then calculate similarly the contribution ##\vec{E_2}## to the electric field from the ring to the right, and then take the sum $$\vec{E}=\vec{E_1}+\vec{E_2}$$ for the total electric field E at point A.
 
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  • #4
Thank you. I'm just wondering, the problem says q1 is greater than q2, and both are positive, so wouldn't I subtract E1-E2? Do I integrate both and then subtract?
 
  • #5
mousey said:
Thank you. I'm just wondering, the problem says q1 is greater than q2, and both are positive, so wouldn't I subtract E1-E2? Do I integrate both and then subtract?
The fields add as vectors, snd as scalars along the x axis. But as scalars, one will have a negative value.
Yes, find each field separately before combining them.
 
  • #6
Yes , well if you find that $$\vec{E_1}=A\hat x$$ where A is a scalar function of ##q_1,R,d## then you ll find that $$\vec{E_2}=-B \hat x$$ where B is another scalar function of ##q_2,R,d##, so that $$\vec{E_1}+\vec{E_2}=(A-B)\hat x$$

Just remember that when you do all the substitutions (for ##\vec{r},\vec{r'} , dr ## e.t.c) you ll end up with an integral that has ##\hat y## and ##\hat z## terms as well as the ##\hat x## term. Just ignore the first two terms (or maybe you can integrate as an excercise to prove that those integrals would be zero) and focus on the ##\hat x## term that is the x-component of the E-field.

And actually the way I 've setup ##\vec{r'}## is for the integral of the upper half of the ring. To calculate the integral for the lower half you got to set ##\vec{r'}=-\sqrt{R^2-z^2}\hat y+z\hat z##. This also means that when you calculate the x-component you got to multiply it by 2 (the x-component is the same for the upper half and for the lower half).
 
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FAQ: Expression for the magnitude of an electric field

What is the expression for the magnitude of an electric field?

The expression for the magnitude of an electric field is given by E = kQ/r^2, where k is the Coulomb's constant, Q is the source charge, and r is the distance from the source charge to the point where the electric field is being measured.

How is the expression for the magnitude of an electric field derived?

The expression for the magnitude of an electric field is derived from Coulomb's Law, which states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. By setting the force equal to the mass times acceleration, and solving for the electric field, we arrive at the expression E = kQ/r^2.

What is the significance of the Coulomb's constant in the expression for the magnitude of an electric field?

The Coulomb's constant, denoted by k, is a proportionality constant that relates the strength of the electric field to the source charge and the distance between the charges. Its value is approximately 8.99 x 10^9 Nm^2/C^2. Without this constant, the expression for the electric field would not accurately reflect the relationship between the source charge and the strength of the electric field.

Can the expression for the magnitude of an electric field be used for both positive and negative charges?

Yes, the expression for the magnitude of an electric field can be used for both positive and negative charges. The direction of the electric field will depend on the sign of the charge, with positive charges producing an outward electric field and negative charges producing an inward electric field.

How is the expression for the magnitude of an electric field related to the concept of electric potential?

The electric field is related to the electric potential by the equation E = -∇V, where E is the electric field, V is the electric potential, and ∇ is the gradient operator. This relationship shows that the electric field is a measure of the change in electric potential over a given distance. The stronger the electric field, the greater the change in electric potential over a given distance.

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