Calculate potential difference from electric field using a curve x=2
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Given an electric field EE, an electric potential VV for EE is any scalar function VV for which
E=−∇VE=−∇V
It follows that if VV is such a potential, then we can integrate both sides along a curve CC to obtain
∫CE⋅dℓ=−∫C∇V⋅dℓ∫CE⋅dℓ=−∫C∇V⋅dℓ
If CC is a curve with endpoints aa and bb, then the gradient theorem tells us that the right hand side can be evaluated in terms of the values of VVat these endpoints alone;
−∫C∇V⋅dℓ=V(a)−V(b)−∫C∇V⋅dℓ=V(a)−V(b)
We now have the freedom to choose a reference point at which we decide what the value of the potential is (this comes from the fact that in step 1, the condition that the field be the gradient of the potential does not uniquely specify the potential) at some chosen reference point b=x0b=x0, and then combining steps 2 and 3 allows us to compute the value of the potential at any other point a=xa=x. In other words, combining these remarks with steps 2 and 3 we obtain
V(x)=V(x0)+∫CE⋅dℓV(x)=V(x0)+∫CE⋅dℓ
where CC is any path from xx to x0x0.
In short, the electric potential is computed by choosing its value at a certain reference point, and then performing a line integral along any path to another point at which you want to determine its value. In this way, you can obtain the functional form of VV at any point xx you like.
E=−∇VE=−∇V
It follows that if VV is such a potential, then we can integrate both sides along a curve CC to obtain
∫CE⋅dℓ=−∫C∇V⋅dℓ∫CE⋅dℓ=−∫C∇V⋅dℓ
If CC is a curve with endpoints aa and bb, then the gradient theorem tells us that the right hand side can be evaluated in terms of the values of VVat these endpoints alone;
−∫C∇V⋅dℓ=V(a)−V(b)−∫C∇V⋅dℓ=V(a)−V(b)
We now have the freedom to choose a reference point at which we decide what the value of the potential is (this comes from the fact that in step 1, the condition that the field be the gradient of the potential does not uniquely specify the potential) at some chosen reference point b=x0b=x0, and then combining steps 2 and 3 allows us to compute the value of the potential at any other point a=xa=x. In other words, combining these remarks with steps 2 and 3 we obtain
V(x)=V(x0)+∫CE⋅dℓV(x)=V(x0)+∫CE⋅dℓ
where CC is any path from xx to x0x0.
In short, the electric potential is computed by choosing its value at a certain reference point, and then performing a line integral along any path to another point at which you want to determine its value. In this way, you can obtain the functional form of VV at any point xx you like.
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