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I'm studying the Lagrangian and its applications in electromagnetism. I stumbled across this inconsistency:

The force of a charge moving through a magnetic field is

## F_b = q v \times B ##

If we define B to be in the ## \hat{z} ## direction, this equation can be written as

## F_b = q ( \dot{\rho} \hat{\rho} + \rho \dot{\phi} \hat{\phi} + z \hat{z} ) \times B \hat{z} ##

## F_b = q \dot{\rho} B (-\hat{\phi}) + q \rho \dot{\phi} B \hat{\rho} ##

## F_b = -q \dot{\rho} B \hat{\phi} + q \rho \dot{\phi} B \hat{\rho} ##

According to John Taylor (Classical Mechanics) the ## \hat{\phi} ## term is equal to zero and the only force the charge experiences is

## F_b = q \rho \dot{\phi} B \hat{\rho} ##

Why is this?