# Proving the B field in a wire only has theta component?

• applestrudle
In summary, the conversation discusses the proof that in a current carrying wire, the magnetic field only has a theta component. This is based on the second and fourth Maxwell equations, which state that the divergence of the magnetic field is zero and that the curl of the magnetic field is equal to the current density multiplied by the permeability of free space. It is also noted that due to cylindrical symmetry, the magnetic field is only dependent on the distance from the z-axis, resulting in equations for the components of the magnetic field in terms of the radial distance. The conversation concludes with the statement that a system of differential equations will need to be solved to fully prove this concept.

## Homework Statement

Prove in a current carrying wire the magnetic field only has a theta component.

## Homework Equations

∇ ⋅ B = 0 (dive of magnetic field zero, 2nd Maxwell Eq)

∇ x B = μ J (Ampere's Law, 4th Maxwell Eq)

Cylindrical symmetry means B field only dependent on r (distance from z axis) so that

B = Br(r)rhat + Bθ(r)θhat + Bz(r)zhat

## The Attempt at a Solution

Divergence of Ampere's Law = 0 gives

- ∂B/∂r θhat + 1/r ∂/∂r(rBθ) zhat = 0

Not really sure where to go from there =/

You'll end up having to solve a system of DEs.

Note: you can write the equations better in LaTeX... i.e. $$\vec B = B_r\hat r + B_\theta\hat\theta + B_\phi\hat\phi \\ -\frac{\partial B}{\partial r}\hat\theta + \frac{1}{r}\frac{\partial}{\partial r}\left(rB_\theta\right) \hat z = 0$$ ... If I understand you correctly that the last equation is supposed to be ##\nabla\cdot(\nabla\times\vec B) = 0## then that does not look right to me.

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## 1. What is the B field in a wire?

The B field, also known as the magnetic field, is a vector field that describes the strength and direction of the magnetic force experienced by a charged particle moving through space.

## 2. How is the B field in a wire measured?

The B field in a wire can be measured using a device called a magnetometer, which detects the strength and direction of the magnetic field by sensing the movement of charged particles.

## 3. How is the B field in a wire affected by current?

The B field in a wire is directly proportional to the amount of electric current flowing through the wire. As the current increases, the strength of the magnetic field also increases.

## 4. What is the direction of the B field in a wire?

The direction of the B field in a wire is determined by the right-hand rule, where the thumb points in the direction of the current flow and the fingers curl in the direction of the magnetic field.

## 5. Why does the B field in a wire only have a theta component?

The B field in a wire only has a theta component because the wire is assumed to be infinitely long and straight, resulting in a uniform magnetic field that is perpendicular to the wire at all points and only varies in magnitude.