- #1

Hamiltonian

- 296

- 193

- Homework Statement
- Calculate the magnitude of the field due to a dipole at a point P which is at a distance ##r## from the midpoint of the two charges and makes an angle ##\theta## with the dipole moment Vector ##P##. the distances between the two opposite charges is ##a## and ##(a<<r)##.

- Relevant Equations
- -

I know how to derive field using ##E = -\nabla V## in polar coordinates and doing so gave me $$E = (kP/r^3)(1 + 3cos^3\theta)^{1/2}$$

now I am trying to derive ##E## at point P using the fields produced by +ve and -ve charge respectively and taking components of each along the radial direction along ##r## and perpendicular to ##r##. I assumed angles ##\alpha## and ##\beta## with the perpendicular to the radial direction in hopes they get eliminated when finding ##E_{net}##.

$$E_+ = \frac {kq}{r^2} (1+ (2a/r) cos\theta)$$

$$E_- = \frac {kq}{r^2} (1- (2a/r) cos\theta)$$

here ##E_+## and ##E_-## are fields due to the -ve and +ve charges at point P.

$$E_r = E_+ sin\beta - E_- simn\alpha$$

$$E_{r'} = E_+ cos\beta + E_- cos\alpha$$

here ##E_r## is the componetnt of ##E_{net}## at P along ##r## and ##E_{r'}## is the component of ##E_{net}## perpendicular to ##r##.

$$| E_{net}| = \sqrt{(E_r)^2 + (E_{r'})^2}$$

I am not able to eliminate ##\alpha## and ##\beta## from the final expression for ##E_{net}##.

now I am trying to derive ##E## at point P using the fields produced by +ve and -ve charge respectively and taking components of each along the radial direction along ##r## and perpendicular to ##r##. I assumed angles ##\alpha## and ##\beta## with the perpendicular to the radial direction in hopes they get eliminated when finding ##E_{net}##.

$$E_+ = \frac {kq}{r^2} (1+ (2a/r) cos\theta)$$

$$E_- = \frac {kq}{r^2} (1- (2a/r) cos\theta)$$

here ##E_+## and ##E_-## are fields due to the -ve and +ve charges at point P.

$$E_r = E_+ sin\beta - E_- simn\alpha$$

$$E_{r'} = E_+ cos\beta + E_- cos\alpha$$

here ##E_r## is the componetnt of ##E_{net}## at P along ##r## and ##E_{r'}## is the component of ##E_{net}## perpendicular to ##r##.

$$| E_{net}| = \sqrt{(E_r)^2 + (E_{r'})^2}$$

I am not able to eliminate ##\alpha## and ##\beta## from the final expression for ##E_{net}##.

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