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Electric Potential

Learn how to calculate electric potential and voltage for continuous charge distributions. Solve problems involving rings and lines of charge. Understand the concept of potential gradient and find electric field from potential expressions.

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Electric Potential

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  1. Electric Potential Voltage

  2. Potential of a Continuous Charge • Break the charge into small dq pieces and find the potential due to each piece, treating it as a point charge • Integrate to find the potential of the whole

  3. Example • A rod of length L has a uniform linear charge density λ. Determine the potential at a point P on the axis of the rod a distance d from one end.

  4. dq = λ dx • For which dV = kdq/r • dV = kλdx/(d + x) • V =∫ol(k λdx/(d + x) • V = kλ∫ol[dx/(d+x)] L d dx P

  5. Continued • ∫dx/(ax + b) • ∫du/u Let u = d +x so du = 1 • ∫dx/(d + x) from 0 to L • ln(d + x) from 0 to L • V = kλ ln[(d+L)/d]

  6. Problem • Find the electric potential at P on the central axis of the ring-shaped charge distribution of net charge Q.

  7. Ring of charge • Consider dq • r = √(x2 + R2) • V = k∫dq/r • V=k∫dq/(√(x2+R2) • V=k/√(x2 + R2)∫dq • V = kQ/√(x2 + R2) dq r R P

  8. Problem • A line of charge Q is distributed uniformly along a line of length 2a. Find the potential at a point P along the perpendicular bisector of the rod at a distance x from its center.

  9. Picture dq √(x2 + y2) P

  10. dq = (Q/2a) dy • V = (kQ/2a) (∫dy/√(x2 + y2)) • From –a to a • V = kQ/2a ln {√ [(a2 + y2) + a]/[√(a2 + x2) – a] }

  11. Potential Gradient • -dV = E dl • E = Exi + Eyj + Ezk • -dV = Exdx + Eydy + Ezdz • Partial Derivative • Ex = -dV/dx • Ey = -dV/dy • Ez = -dV/dz • All partial derivatives

  12. Gradient of Voltage • E = -(i dV/dx +j dV/dy + k dV/dz) • E = - V • E is the – Gradient of V

  13. Problem • From the potential at a radial distance from a point charge q V = kq/r. Find the vector electric field from this expression for V.

  14. Answer • Er = -dV/dr = -d/dr (kq/r) • Er = kq/r2

  15. Problem • We found that for a ring of charge with a radius a and total charge Q, the potential at a point P on a ring axis a distance x from its center is • V = kQ/(√x2 +a2)

  16. Problem • V = (6.00 V/m) x + (4.00 V/m) y2 + (0.00 V/m) z • Find E.

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