Concept explainers
An electric dipole is located along the y axis as shown in Figure P24.48. The magnitude of its electric dipole moment is defined as p = 2aq. (a) At a point P, which is far from the dipole (r ≫ a), show that the electric potential is
(b) Calculate the radial component Er and the perpendicular component Eθ of the associated electric field. Note that Eθ = −(1/r)(∂V/∂θ). Do these results seem reasonable for (c) θ = 90° and 0°? (d) For r = 0? (e) For the dipole arrangement shown in Figure P24.48, express V in terms of Cartesian coordinates using r = (x2 + y2)1/2 and
(f) Using these results and again taking r ≫ a, calculate the field components Ex and Ey.
Figure P24.48
(a)
To show: The electric potential at a point P is
Answer to Problem 48CP
The charge on the insulating sphere is
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The expression to calculate the total electric potential of the dipole is,
Since,
Substitute
Substitute
Conclusion:
Therefore, the electric potential at a point P is
(b)
The radial component
Answer to Problem 48CP
The radial component is
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
Write the expression of radial component
Substitute
Write the expression to calculate the and perpendicular component
Substitute
Thus, the radial component is
Conclusion:
Therefore, radial component is
(c)
Whether the values of
Answer to Problem 48CP
These values of radial component and perpendicular component of electric field seem reasonable.
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The radial component of electric filed at point P is,
Substitute
Substitute
The perpendicular component of electric field at point P is,
Substitute
Substitute
Thus, these values of radial component and perpendicular component of electric field seem reasonable.
Conclusion:
Therefore, these values of radial component and perpendicular component of electric field seem reasonable.
(d)
Whether the values of
Answer to Problem 48CP
The value of radial component and perpendicular component of electric field at
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The magnitude of the electric field between the charges of the dipole is not infinite.
The radial component of electric filed at point P is,
The perpendicular component of electric field at point P is,
The values of
Conclusion:
Therefore, the value of radial component and perpendicular component of electric field at
(e)
The electric potential of dipole in terms of Cartesian coordinates.
Answer to Problem 48CP
The electric potential of dipole in terms of Cartesian coordinates is
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The electric potential at a point P is,
If
Substitute
Conclusion:
Therefore, the electric potential of dipole in terms of Cartesian coordinates is
(e)
The field component
Answer to Problem 48CP
The field component
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The x component of the electric field is,
Substitute
The y component of the electric field is,
Substitute
Conclusion:
Therefore, the field component
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