8.9: Magnetism from Electricity

Learning Objectives

• Calculate current that produces a magnetic field.
• Use the right hand rule 2 to determine the direction of current or the direction of magnetic field loops.

How much current is needed to produce a significant magnetic field, perhaps as strong as the Earth’s field? Surveyors will tell you that overhead electric power lines create magnetic fields that interfere with their compass readings. Indeed, when Oersted discovered in 1820 that a current in a wire affected a compass needle, he was not dealing with extremely large currents. How does the shape of wires carrying current affect the shape of the magnetic field created? We noted earlier that a current loop created a magnetic field similar to that of a bar magnet, but what about a straight wire or a toroid (doughnut)? How is the direction of a current-created field related to the direction of the current? Answers to these questions are explored in this section, together with a brief discussion of the law governing the fields created by currents.

Magnetic Field Created by a Long Straight Current-Carrying Wire

If we shift our perspective to consider the effect that the moving charge has on the magnetic field, as in Figure \(\PageIndex{1}\), we can see how electrical current can produce a magnetic effect. Please note, this is the exact same relationship we were looking at previously, we are simply shifting the perspective to the relationship between electrical current and magnetic field instead of electrical current and magnetic force.

Figure \(\PageIndex{1}\): (a) Compasses placed near a long straight current-carrying wire indicate that field lines form circular loops centered on the wire. (b) Right hand rule 2 states that, if the right hand thumb points in the direction of the current, the fingers curl in the direction of the field. This rule is consistent with the field mapped for the long straight wire and is valid for any current segment.

Ampere’s Law and Others

The magnetic field of a long straight wire has more implications than you might at first suspect. Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment. The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law. Using advanced mathematical techniques that are beyond the scope of this course results in a more complete law, Ampere’s law. This law relates magnetic field and current in a general for a variety of shapes for both the wire and the field. Leaving aside the math, Ampere's Law can be stated as:

Ampere's Law

The magnetic field created by an electrical current is proportional to this electrical current.

More generally it can be stated as magnetic effects can be produced by electrical phenomenon.

Magnetic Field Produced by a Current-Carrying Circular Loop

The magnetic field near a current-carrying loop of wire is shown in Figure \(\PageIndex{2}\). Consider how the direction of the electrical current and the magnetic field are related.

Figure \(\PageIndex{2}\): (a) RHR-2 gives the direction of the magnetic field inside and outside a current-carrying loop. (b) More detailed mapping with compasses or with a Hall probe completes the picture. The field is similar to that of a bar magnet.

Magnetic Field Produced by a Current-Carrying Solenoid

A solenoid is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero. Figure \(\PageIndex{3}\) shows how the field looks. The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the magnetic field strength inside a solenoid is related to the number of loops and the current in the wire.

Figure \(\PageIndex{3}\): (a) Because of its shape, the field inside a solenoid of length \(l\) is remarkably uniform in magnitude and direction, as indicated by the straight and uniformly spaced field lines. The field outside the coils is nearly zero. (b) This cutaway shows the magnetic field generated by the current in the solenoid.

There are interesting variations of the flat coil and solenoid. For example, the toroidal coil used to confine the reactive particles in tokamaks is much like a solenoid bent into a circle. The field inside a toroid is very strong but circular. Charged particles travel in circles, following the field lines, and collide with one another, perhaps inducing fusion. But the charged particles do not cross field lines and escape the toroid. A whole range of coil shapes are used to produce all sorts of magnetic field shapes. Adding ferromagnetic materials produces greater field strengths and can have a significant effect on the shape of the field. Ferromagnetic materials tend to trap magnetic fields (the field lines bend into the ferromagnetic material, leaving weaker fields outside it) and are used as shields for devices that are adversely affected by magnetic fields, including the Earth’s magnetic field.

Section Summary

• The magnetic field is perpendicular to the direction of the electric current which generated it.
• The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampere’s law.
• The magnetic field strength created by an electrical current is proportional to this electrical current.
• The magnetic field strength inside a solenoid is proportional the the current and the number of loops of wire. The field inside is very uniform in magnitude and direction.

Glossary

solenoid

a thin wire wound into a coil that produces a magnetic field when an electric current is passed through it

Biot-Savart law

a physical law that describes the magnetic field generated by an electric current in terms of a specific equation

Ampere’s law

the physical law that states that the magnetic field around an electric current is proportional to the current; each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment