Solution The number of turns per unit length is. Significance This solution is valid only if the length of the solenoid is reasonably large compared with its diameter. This example is a case where this is valid.
Check Your Understanding What is the ratio of the magnetic field produced from using a finite formula over the infinite approximation for an angle of a b The solenoid has turns in 50 cm with a current of 1. A toroid is a donut-shaped coil closely wound with one continuous wire, as illustrated in part a of Figure. If the toroid has N windings and the current in the wire is I , what is the magnetic field both inside and outside the toroid?
However, if the toroid is tightly wound, all points on the circle become essentially equivalent [part c of Figure ], and cylindrical symmetry is an accurate approximation.
This allows us to write for each of the paths and shown in part d of Figure ,. For a path that is external to the toroid, either no current passes through the enclosing surface path , or the current passing through the surface in one direction is exactly balanced by the current passing through it in the opposite direction In either case, there is no net current passing through the surface, so.
The turns of a toroid form a helix, rather than circular loops. As a result, there is a small field external to the coil; however, the derivation above holds if the coils were circular.
For a circular path within the toroid path , the current in the wire cuts the surface N times, resulting in a net current NI through the surface. The magnetic field is directed in the counterclockwise direction for the windings shown.
When the current in the coils is reversed, the direction of the magnetic field also reverses. However, if the central radius R the radius midway between the inner and outer radii of the toroid is much larger than the cross-sectional diameter of the coils r , the variation is fairly small, and the magnitude of the magnetic field may be calculated by Figure where.
The field inside is very uniform in magnitude and direction. The magnetic field strength inside a toroid is where N is the number of windings. Conceptual Questions Is the magnetic field inside a toroid completely uniform? Almost uniform? Explain why inside a long, hollow copper pipe that is carrying an electric current parallel to the axis. Is outside the pipe? Outside the pipe, there may be an enclosed current through the copper pipe, so the magnetic field may not be zero outside the pipe.
A solenoid is wound with turns per meter. When the current is 5. A solenoid has 12 turns per centimeter. What current will produce a magnetic field of within the solenoid?
If a current is 2. A solenoid is 40 cm long, has a diameter of 3. If the current through the windings is 4. In engineering , the term solenoid may also refer to a variety of transducer devices that convert energy into linear motion. The term is also often used to refer to a solenoid valve , which is an integrated device containing an electromechanical solenoid which actuates either a pneumatic or hydraulic valve, or a solenoid switch, which is a specific type of relay that internally uses an electromechanical solenoid to operate an electrical switch; for example, an automobile starter solenoid , or a linear solenoid, which is an electromechanical solenoid.
This is a derivation of the magnetic field around a solenoid that is long enough so that fringe effects can be ignored.
In the diagram to the right, we immediately know that the field points in the positive z direction inside the solenoid, and in the negative z direction outside the solenoid. We see this by applying the right hand grip rule for the field around a wire. If we wrap our right hand around a wire with the thumb pointing in the direction of the current, the curl of the fingers shows how the field behaves.
Since we are dealing with a long solenoid, all of the components of the magnetic field not pointing upwards cancel out by symmetry. Outside, a similar cancellation occurs, and the field is only pointing downwards. Now consider imaginary the loop c that is located inside the solenoid. We have shown above that the field is pointing upwards inside the solenoid, so the horizontal portions of loop c doesn't contribute anything to the integral. Thus the integral of the up side 1 is equal to the integral of the down side 2.
Inside the toroid, the magnetic field forms concentric circles not shown. Since the magnetic field is parallel to the Amperial loop everywhere along the loop, and the magnetic field does not change magnitude by symmetry , the circulation is given by:. It is easy to show, by using Amperian loops that are either smaller or bigger than the toroid, that the magnetic field everywhere outside of the toroid is exactly zero as those Amperian loops will enclose no net current.
In a toroid, the magnetic field lines form closed circles.
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