What Is the Interpretation of an Integral Equation with a Proportional Variable?

[Anti Hydrogen]

Homework Statement

Hi, I'm trying to understand the case when a variable is proportional to that which is under a integral sign, for example
$$\Delta V = - \int_a^b \vec E \cdot d \vec s$$

What could be a interpretation of this equation?

Relevant Equations

Thanks

Thanks

 

[PeroK]

Homework Statement:: Hi, I'm trying to understand the case when a variable is proportional to that which is under a integral sign, for example
$$\Delta V = - \int_a^b \vec E \cdot d \vec s$$

What could be a interpretation of this equation?
Relevant Equations:: Thanks

Thanks

An integral is the limit of a summation process. Here we have $\Delta V$ as (the sum of) the product of the electric field multiplied by a length. You could compare this with:
$$W = \int_a^b \vec F \cdot d \vec r$$

 

[etotheipi]

Essentially this is a restatement what @PeroK was saying: consider the change in the potential $\delta V$ along a small displacement vector $\delta \vec{r}$ in a region where the electric field is $\vec{E}$, $$\delta V = -\vec{E} \cdot \delta \vec{r}$$Now if we sum up all of these small changes along some path $$\sum \delta V = -\sum \vec{E} \cdot \delta \vec{r}$$If you now take the limit as the length of the displacement vector approaches zero, that is $\delta \vec{r} \rightarrow d\vec{r}$ and consequently also $\delta V \rightarrow dV$, the sums turn into integrals $$\int_{V_{1}}^{V_{2}} dV = -\int_{a}^{b} \vec{E} \cdot d\vec{r}$$Apologies if I abused any notation! For this specific integral, the interpretation is the change in potential going from $a$ to $b$.

 

[kuruman]

Homework Statement:: Hi, I'm trying to understand the case when a variable is proportional to that which is under a integral sign, for example
$$\Delta V = - \int_a^b \vec E \cdot d \vec s$$

What could be a interpretation of this equation?
Relevant Equations:: Thanks

Thanks

What variable is proportional to what under the integral sign? Are you saying that $\Delta V$ is proportional to $\vec E \cdot d \vec s?$ @PeroK said it right, look at the line integral as a summation.