The Electric Field — Question 3

Question 3

Rotating the coordinate axes changes components but not the field

An electric field has components $(E_x,E_y)=(60,80)$ N/C. New axes are rotated counterclockwise by $\phi=30^\circ$, with $\hat{\mathbf x}\prime=\cos\phi\,\hat{\mathbf x}+\sin\phi\,\hat{\mathbf y}$.

1. Find the field components in the rotated axes by projection.

2. Check the magnitude and express the force on a $-2.00$ nC probe in the rotated basis.

Question 3 - Solution

Rotating the coordinate axes changes components but not the field

(a) Method and calculation

$$E_x\prime=E_x\cos\phi+E_y\sin\phi=30\sqrt3+40\approx91.9615\,\mathrm{N/C}.$$$$E_y\prime=-E_x\sin\phi+E_y\cos\phi=-30+40\sqrt3\approx39.2820\,\mathrm{N/C}.$$

(b) Interpretation and further calculation

$$(E_x\prime)^2+(E_y\prime)^2=60^2+80^2=10^4\,\mathrm{(N/C)^2}.$$$$\mathbf F\prime=(-183.923\hat{\mathbf x}\prime-78.5641\hat{\mathbf y}\prime)\,\mathrm{nN}.$$The vector has not physically rotated; only its numerical description has.

Physical check

An orthonormal rotation preserves dot products and therefore the squared magnitude.