Magnetic Field of a Current-Carrying Wire

Direction of Magnetic Field around a Current-Carrying Wire

So how can we determine the direction of the generated magnetic field? We can use something called the right-hand grip rule. Referring to the figure below, we orient your right hand with the thumb pointing upwards whilst the fingers are curved towards your palm. In this case, your thumb is pointed in the direction of the current whilst the fingers are curved in the direction of the magnetic field.

Let's consider an example.

Magnetic Field Strength of a Current-Carrying Wire

Now let's see how the magnetic field strength of a current-carrying wire affects other objects, for instance, another wire.

Magnetic Field Strength between Two Current-Carrying Wires

Now we have established how to calculate the magnetic field generated by the current in a wire, let's look at how these magnetic fields may interact with one another. If you have read our other article on the magnetic field of a moving charge, you may be familiar with how moving charges interact with magnetic fields via the Lorentz force. If not, that's okay, here's a quick refresher!

When charged particles travel through a magnetic field, they experience something called the Lorentz force, which is given by

\[ \vec{F}_{\text{M}} = q \vec{v} \times \vec{B},\]

where \(\vec{F}_{\text{M}}\) is the Lorentz force measured in newtons \(\mathrm{N}\), \(q\) is the charge of the particle measured in coulombs \(\mathrm{C}\), \(\vec{v}\) is the velocity of the charged particle measured in \(\mathrm{\frac{m}{s}}\), and \(\vec{B}\) is the magnetic field measured in Teslas \(\mathrm{T}\). Furthermore, we can also represent the magnitude of the Lorentz force as

\[ |\vec{F}_{\text{M}}| = |q\vec{v}| |\sin(\theta)| |\vec{B}| ,\]

where \(\theta\) is the angle between the trajectory of the charged particle and the magnetic field \(B\), and all the other quantities are the same as above.

The magnitude of the Lorentz force on a wire is given as

\[ |\vec{F}_{\text{M}}| = |I \vec{L}| |\sin(\theta) | |\vec{B} | ,\]

where \(I\) is the current running in the wire measured in \(\mathrm{A}\), \(L\) is the length of the wire in the magnetic field measured in \(\mathrm{m}\), and all the other quantities are the same as before. The direction of the resultant Lorentz force can be determined using the left-hand rule.

Derivation of Lorentz Force on a Wire

This derivation is a deep dive and is not needed in the scope of the course.

Right-Hand Rule to find the Magnetic Field of a Current-Carrying Wire

Now let's consider two wires running parallel to one another as in the figure below.

Fig. 4 - Two parallel wires exerting a force on one another.

Consider the force \(F_1\) acting on the wire on the right-hand side carrying current \(I_2\). The magnetic field generated from the wire on the left is acting downwards on the wire on the right, whilst the current is running into the page. If we use our left-hand rule and orient the first finger downwards, whilst the second finger points into the page, we find that the resultant force \(F_1\) (given by the direction of the thumb) is pointing towards the left.

Carrying out a similar process with the wire on the left, this time instead of the magnetic field from the wire on the right acting upwards, we find that the resultant Lorentz force \(F_2\) is pointing towards the right. Thus we can conclude that two wires carrying currents in the same direction will experience an attractive force toward one another.

Can you find the direction of the force if the currents are running in opposite directions? Look to our flashcards for this article to test yourself!

Formula of the Magnetic Force between Two Current-Carrying Wires

Consider again the situation above where the two wires are carrying currents in the same direction. Using the magnitude of the Lorentz force equation, we can substitute our expression for the magnetic field from wire 1, to find the force on wire 2 as

\[ \begin{align} F &= BI_2 L\sin(\theta) \\ F &= \frac{\mu_0}{2\pi}\frac{I_1}{r} I_2 L \sin(90^{\circ}) \\ F &= \frac{\mu_0}{2\pi} \frac{I_1 I_2}{r} L . \end{align} \]

Here the first line of our derivation indicates the force on the second wire, which is why we have replaced the \(I\) term with \(I_2\). Then, we substitute the expression of the magnetic field from the first wire, which results in the inclusion of the \(I_1\) term.

Eliminating the trigonometry term as that goes to one, we can rearrange to find the force per unit length on the second wire as

\[ \frac{F}{L} = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{r} , \]

where \(\frac{F}{L}\) is the force per unit meter measured in newtons per meter \(\mathrm{N}{m}\), \(r\) is the radial distance from the second wire, and all the other terms are the same as previous equations.

Magnetic Field due to Current-Carrying Circular Loop

We now look at a specific case of the shape of a current-carrying wire; in the case of the wire forming a loop, our formula for the magnetic field strength around the wire changes slightly. In this instance, our equation becomes

\[ B = \frac{\mu_0 I}{2R} ,\]

where \(B\) is the magnitude of the magnetic field strength at the center of the loop measured in teslas \(\mathrm{T}\), \(R\) is the radius of the current loop, and all the other quantities remain the same as our previous equation.

Thus, we can see that the formula looks very similar to the general magnetic field equation around a wire, but some small differences need to be memorized. For instance, we no longer have the factor of \(2\pi\) on the denominator as the shape of the surrounding magnetic field is no longer in circular rings. Additionally, the radius \(R\) is no longer a variable but rather a constant.

Magnetic Force on Current-Carrying Wires Example

Finally, let's consider an example question.