How Does the Electric Field Influence Potential Difference Calculation?

[Arman777]

Homework Statement

##V = V (x,y,z) = 10^3(2x^2 + 3y^2 −5z^2 + 7x)−19999##
Find ##V_{ba}
##\vec {r_a}=2i##
##\vec {r_b}=j+k##

Homework Equations

##E## will be ##E=-∇V##
##V_b-V_a=-\int_a^b E(r)\, dr##

The Attempt at a Solution

When I try to calculate the potential using ##V_b-V_a=-\int_a^b E(r)dr##, and ##E=-∇V## dr I get ##-26.10^3##
Electric field at a is ##-15.10^3i##
Electrci Field at b is ##-6.10^3j+10.10^3k##
so inital ##V## is ##-30.10^3 V## final ##V## is ##-4.10^3 V## from integral and thres minus sign so its ##-26.10^3 V##
when I use ##V = V (x,y,z) = 10^3(2x^2 + 3y^2 −5z^2 + 7x)−19999## inserting here I get ##-24.10^3V##

 

[TSny]

##V_b-V_a=-\int_a^b E(r)dr##

The general expression is ##V_b-V_a=-\int_a^b \vec{E} \cdot d\vec{s}##, where the integration is along some path connecting the points a and b. The integrand involves a dot product.

Electric field at a is ##-15.10^3i##
Electrci Field at b is ##-6.10^3j+10.10^3k##
so inital ##V## is ##-30.10^3 V## final ##V## is ##-4.10^3 V## from integral and thres minus sign so its ##-26.10^3 V##

It's not clear how you did the integration. What path did you choose? Why did you evaluate ##\vec E## at the specific points a and b? The integral depends on ##\vec E## at all points of the path of integration.