The electric field in the given system is expressed as vec E = (10x^2 + 7x), where x represents the position vector in a three-dimensional space. We need to calculate the potential difference at point H, which is located at a distance of 3 m from the origin.

To calculate the potential difference, we first need to find the electric potential function. The electric potential (V) at any point in space is given by the negative gradient of the electric potential energy (U). Mathematically, it can be expressed as:

vec E = -∇V

Here, ∇ represents the gradient operator. In three-dimensional Cartesian coordinates (x, y, z), the gradient operator can be written as:

∇ = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k

where i, j, and k are the unit vectors along the x, y, and z directions, respectively.

Therefore, applying the gradient operator on the electric potential function, we have:

∇V = (∂V/∂x)i + (∂V/∂y)j + (∂V/∂z)k

Comparing this with the given electric field expression, we can equate the coefficients of the unit vectors to find the partial derivatives of V with respect to x, y, and z.

Since the given electric field is only a function of x, we can conclude that the electric potential is also only a function of x. Thus, (∂V/∂y) and (∂V/∂z) are both zero.

Therefore, we have:

(∂V/∂x)i = (10x^2 + 7x)i

Integrating both sides with respect to x, we get:

V = ∫(10x^2 + 7x)dx

Calculating the integral, we have:

V = (10/3)x^3 + (7/2)x^2 + C

where C is the constant of integration.

Now, to calculate the potential difference (ΔV) at point H, we can subtract the electric potential at the origin (V = 0) from the electric potential at point H (V = (10/3)(3^3) + (7/2)(3^2) + C):

ΔV = (10/3)(27) + (7/2)(9) + C

Simplifying this expression, we can calculate the potential difference at point H.

In summary, the electric field in the given system is (10x^2 + 7x), and to find the potential difference at point H, we need to calculate the electric potential function using the negative gradient of the electric potential energy. By integrating the given electric field expression, we can find the electric potential function and then calculate the potential difference at point H by subtracting the electric potential at the origin.