<Dd> m x _̈ + q (∂ A x ∂ t + ∂ A x ∂ x x _̇ + ∂ A x ∂ y y _̇ + ∂ A x ∂ z z _̇) = − q ∂ φ ∂ x + q (∂ A x ∂ x x _̇ + ∂ A y ∂ x y _̇ + ∂ A z ∂ x z _̇) (\ displaystyle m (\ ddot (x)) + q \ left ((\ frac (\ partial A_ (x)) (\ partial t)) + (\ frac (\ partial A_ (x)) (\ partial x)) (\ dot (x)) + (\ frac (\ partial A_ (x)) (\ partial y)) (\ dot (y)) + (\ frac (\ partial A_ (x)) (\ partial z)) (\ dot (z)) \ right) = - q (\ frac (\ partial \ phi) (\ partial x)) + q \ left ((\ frac (\ partial A_ (x)) (\ partial x)) (\ dot (x)) + (\ frac (\ partial A_ (y)) (\ partial x)) (\ dot (y)) + (\ frac (\ partial A_ (z)) (\ partial x)) (\ dot (z)) \ right)) </Dd> <Dl> <Dd> F x = − q (∂ φ ∂ x + ∂ A x ∂ t) + q (y _̇ (∂ A y ∂ x − ∂ A x ∂ y) + z _̇ (∂ A z ∂ x − ∂ A x ∂ z)) = q E x + q (y _̇ (∇ × A) z − z _̇ (∇ × A) y) = q E x + q (r _̇ × (∇ × A)) x = q E x + q (r _̇ × B) x (\ displaystyle (\ begin (aligned) F_ (x) & = - q \ left ((\ frac (\ partial \ phi) (\ partial x)) + (\ frac (\ partial A_ (x)) (\ partial t)) \ right) + q \ left ((\ dot (y)) \ left ((\ frac (\ partial A_ (y)) (\ partial x)) - (\ frac (\ partial A_ (x)) (\ partial y)) \ right) + (\ dot (z)) \ left ((\ frac (\ partial A_ (z)) (\ partial x)) - (\ frac (\ partial A_ (x)) (\ partial z)) \ right) \ right) \ \ & = qE_ (x) + q ((\ dot (y)) (\ nabla \ times \ mathbf (A)) _ (z) - (\ dot (z)) (\ nabla \ times \ mathbf (A)) _ (y)) \ \ & = qE_ (x) + q (\ mathbf (\ dot (r)) \ times (\ nabla \ times \ mathbf (A))) _ (x) \ \ & = qE_ (x) + q (\ mathbf (\ dot (r)) \ times \ mathbf (B)) _ (x) \ end (aligned))) </Dd> </Dl> <Dd> F x = − q (∂ φ ∂ x + ∂ A x ∂ t) + q (y _̇ (∂ A y ∂ x − ∂ A x ∂ y) + z _̇ (∂ A z ∂ x − ∂ A x ∂ z)) = q E x + q (y _̇ (∇ × A) z − z _̇ (∇ × A) y) = q E x + q (r _̇ × (∇ × A)) x = q E x + q (r _̇ × B) x (\ displaystyle (\ begin (aligned) F_ (x) & = - q \ left ((\ frac (\ partial \ phi) (\ partial x)) + (\ frac (\ partial A_ (x)) (\ partial t)) \ right) + q \ left ((\ dot (y)) \ left ((\ frac (\ partial A_ (y)) (\ partial x)) - (\ frac (\ partial A_ (x)) (\ partial y)) \ right) + (\ dot (z)) \ left ((\ frac (\ partial A_ (z)) (\ partial x)) - (\ frac (\ partial A_ (x)) (\ partial z)) \ right) \ right) \ \ & = qE_ (x) + q ((\ dot (y)) (\ nabla \ times \ mathbf (A)) _ (z) - (\ dot (z)) (\ nabla \ times \ mathbf (A)) _ (y)) \ \ & = qE_ (x) + q (\ mathbf (\ dot (r)) \ times (\ nabla \ times \ mathbf (A))) _ (x) \ \ & = qE_ (x) + q (\ mathbf (\ dot (r)) \ times \ mathbf (B)) _ (x) \ end (aligned))) </Dd> <P> and similarly for the y and z directions . Hence the force equation is: </P>

Velocity of a charge in a magnetic field