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Determine whether the vector field is conservative and, if so, find the general potential function.
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- Given the potential energy function U(x) =−x^2 + 4x+ 2 calculate the force F(x) associated with U(x), and find the maxima/minima of the potential. Does the potential have a point of stability?Find a scalar potential, fi, if F=(3x^2yz^2)i+(x^3z^2)j+(2x^3yz)k, and fi(1,1,1)=1The force F= (yzi + zxj + xyz)/xyz acts on the particle P(x,y,z) which moves in space. (a) Using the relation derived in Prob. 13.79, show that this force is a conservative force. (b) Determine the potential function associated with F.Reference to Problem 13.79:Prove that a force F(x,y,z) is conservative if, and only if, the following relations are satisfied:
- For the scalar potential field f = x^2*y^3 + x*y^2*z^3 + c, what is the direction and magnitude of the steepest decent at point P(1,2,1) and calculate the directional derivative in the direction of u(2,1,2) at the point P?A) The force vector is conservative or non conservative B) The potential function of the force vectorFind the electric potential at a distance h from the center of a disk of radius R, with a charge distribution homogeneous σ
- Could the vector A = x 2 y i + x y 2j + e-βy sinαx k, where α and β are constant, be interpreted as a conservative electric field? If the answer is affirmative, find the potential V from which the electric field could be obtained. (The bold letters indicate vectors. Place the arrows on the vectors and the "hats" on the unit vectors.)Find the electric potential at a distance h from the center of a ring of radius R, with a charge distribution homogeneous λHow did you solve D? Wouldn't it's distance be different from that of Point 1 to Point 2 as would be Point 3 to Point 2 and shouldn't that effect the answer for work needed to be done?
- question:- Consider in three-dimensional space a particle that moves at a central potential . Then the equation of motion of the particle is F=m((d^2)r/dt^2)=-∇U(|r|) Consider the transformation of the right-hand side of the equation, ∇U (r), in the Galileo transformation r ′ = r - vt and state that the equation of motion is a Galileo invariant.The superposition of three potentialsIn a xy plane, we fix a 1uC particle A at the origin and a 2uC particle B at (x=4m; y=0). (a) Calculate the electric potential in (x=4m; y=3m). (b) Where should a C particle of -4uC be placed so that the potential at (x=4m;y=3m) is zero?Find the electric potential inside and outside a uniformly charged spherical shell of radius R