Calculating total charge when the electric field is given

• Saptarshi Sarkar
In summary: The main issue with your computation is that it is not valid for ##r = 0##, which is where the point charge at the origin would contribute. This is why the two answers are different, even though they are both derived from Gauss' law. It's important to make sure your computation method is appropriate for the given scenario.
Saptarshi Sarkar
Homework Statement
Calculate the total charge of an unknown charge distribution for which the electric field is E=q/r^2 e^(-4r) r ̂
Relevant Equations
E.A = q/ε0
∇·E = ρ/ε0
I first tried to use the Gauss' law equation E.A = q/ε0 to find the total charge enclosed. The answer came out to be q(enclosed) = 4πqε0e^(-4r). So for r approaching infinity, q(enclosed) approached 0.

Next, I tried the equation ∇·E = ρ/ε0, calculated rho to be -4qε0e^(-4r)/r^2 and total charge to be -4πqε0.

Why are the answers different although both are derived from Gauss' law? What did I do wrong?

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Your computation for the divergence of the electric field is only valid for ##r > 0##. This means that you will miss the contribution from the point charge at the origin.

Saptarshi Sarkar
Another thing you might think about is how the given electric field behaves as ##r## gets really big. Compare this to what the field of any point charge, ##q_o##, behaves for large ##r##. What does ##q_o## have to equal so that these distant fields are the same.

Saptarshi Sarkar
Paul Colby said:
Another thing you might think about is how the given electric field behaves as ##r## gets really big. Compare this to what the field of any point charge, ##q_o##, behaves for large ##r##. What does ##q_o## have to equal so that these distant fields are the same.

For both the cases, the electric field approaches 0 as r approaches large numbers. Only conclusion I can draw from this is that the given electric field falls to 0 faster due to the exponential term.

Paul Colby
Orodruin said:
Your computation for the divergence of the electric field is only valid for ##r > 0##. This means that you will miss the contribution from the point charge at the origin.

I am not sure I am able to understand. How do I know that there is a charge at the centre and how should I calculate it?

If you compute the divergence with a method valid at ##r = 0##, you will find that, apart from your result, there is a delta function at the origin.

Alternatively, you can check this by taking the flux through a sphere with ##r \to 0##, which will yield a non-zero result, indicating that there is a point charge at the origin.

Saptarshi Sarkar
Yes.

Saptarshi Sarkar

1. How do you calculate total charge when the electric field is given?

To calculate total charge when the electric field is given, you can use the equation Q = E x A, where Q is the total charge, E is the electric field, and A is the area of the surface where the electric field is acting.

2. What units are used when calculating total charge with a given electric field?

The units used when calculating total charge with a given electric field are coulombs (C) for charge, newtons per coulomb (N/C) for electric field, and square meters (m^2) for area. Make sure to use consistent units when plugging in values to the equation Q = E x A.

3. Can total charge be negative when the electric field is given?

Yes, total charge can be negative when the electric field is given. This indicates that the charge is negative and is moving in the opposite direction of the electric field. However, when using the equation Q = E x A, the charge must be represented as a positive value.

4. What is the relationship between electric field and total charge?

The relationship between electric field and total charge is directly proportional. This means that as the electric field increases, the total charge also increases, and vice versa. This relationship is described by the equation Q = E x A, where Q is directly proportional to E.

5. Can the area be a negative value when calculating total charge with a given electric field?

No, the area cannot be a negative value when calculating total charge with a given electric field. This is because the area represents a physical surface, and it cannot have a negative value. If the area is in a direction opposite to the electric field, it should be represented as a negative value in the equation Q = E x A.

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