Magnetic+Fields+Due+To+Currents

Did you know ? = = Magnetic Resonance Imaging (MRI) test can have some potential risks. Severe skin burns have been reported when the patient is tested under anesthesia. 

"//Leads or// [|//wires//] //that are used in the// [|//magnet//] //bore during Magnetic Resonance Imaging (MRI) procedures, should not form large-radius wire loops. Leg-to-leg and leg-to-arm skin contact through wires should be prevented in order to avoid the risk of burning due to the generation of high current loops if the legs or arms are allowed to touch. Physics behind this, is Faraday’s induction Laws. According to the reports, disregards of Faraday’s laws of induction has led to severe skin burns on sedated patients//” = = = = =Magnetic Fields Due to Currents= Moving charged particles produce magnetic fields around themselves. These currents of charged particles produce magnetic fields around the current-carrying element. By combining the study of electric and magnetic fields we come up with the term **electromagnetism.**

Electromagnetism has helped lead to inventions such as computer floppies, video tapes, cassettes, and electric motors. Almost everyone has had a chance in their lives to play with or use magnets, either a permanent magnet such as one that you stick on your fridge or ones that we may not even know exists such as memory in our computers. Electromagnets are used in televisions, motors, and even to diagnose medical conditions.

Below is a picture of Faraday’s first application of electromagnetism; a rod that spun when hung over a battery and current was run through it:

This spinning motion can also be accomplished by some other physics derived later in the page. Here is an example of what two wires with current running through them can do:

media type="youtube" key="dhnnspCLA6Y" height="344" width="425" =Calculating the magnetic field due to a current= If one wanted to calculate the magnetic field produced by the rod in Faraday’s experiment, one could do so using the **Biot-Savart Law.**

The **Biot-Savart Law** is an equation which would describe the magnetic field (**//B//**) created by an electric current (**//i//**) of some object at a point **//r//** distance away. Here is the equation: The **permeability constant** (µ) in this equation is given as: The **Biot-Savart Law** is so flexible because it can be used to find the magnitude of a magnetic field (caused by current) for a variety of current distributions. Each type of distribution can rework the **Biot-Savart Law** to a simpler equation, making it easier to solve for **//B//** (the magnitude of the magnetic force)**//.//**

Magnetic Field Due to a Current in a Long Straight Wire
The first type of distribution of current will be that in a long straight wire. The equation for the magnitude of a magnetic field perpendicular distance **//R//** to this wire (with current **//i//**) is given as:

A similar equation is used when the straight wire is not infinitely long:

The current (**//i//**) travels through the wire in one direction. In order to determine the direction that the magnetic field is applying its force, the **right hand rule** can be applied: Grasp the wire with your right hand, pointing your thumb in the direction that the current is flowing. When you curl your fingers around the wire, the direction your forefingers point is the direction of the magnetic field. If the current were going in the downward direction, the hand in the picture would be upside down and the magnetic field would be in a clockwise direction. =Magnetic Field of a Circular Arc= Finding the magnetic force of a straight wire is pretty straight forward. What about if that wire was curved into an arc? For the following picture we want to find the magnetic force at **C** which is distance **R** from the wire which carries the current **i** over the arc of the angle. To find the magnetic field of a circular arc, we use the **Biot-Savart Law** again and use a bit of calculus in order to get a generalized equation: This sets up the **Biot-Savart Law** equation for a point perpendicular to the arc where we integrate over the distance of the wire (across the arc with a given angle): Solving for B, we end up with the following equation: =Force between Parallel Currents= When two wires carrying current are parallel to each other, they apply their magnetic forces to each other. To find the force on a current-carrying wire due to a second parallel current-carrying wire, first find the field due to the second wire at the site of the first wire. Then find the force on the first wire due to that field. Here is the equation to figure out that force:

To help clarify how this equation would apply, look at the diagram below:

In (a), the wires have opposite currents so their forces repel each other. In (b), the currents are going the same direction; therefore, the forces are attracting each other. =Ampere’s Law= For the below picture we have three wires, two of them are contained in the Amperian loop. When finding the magnitude of the magnetic force we would integrate the contour that surrounds wires 1 and 2, leaving out the third wire. The magnitude would result in the permeability constant multiplied by the total current of wires 1 and 2. The direction of the magnetic force would depend on which wire had the greater current. To determine the current’s direction (positive or negative) we use the right hand rule. Curling your forefingers in the direction of the loop’s integration, the direction your thumb is pointing will give the positive currents and the ones in the opposite direction will be negative currents. See the picture below for a better view: =Fields of a solenoid and Toroids= A **solenoid** is a three-dimensional loop of wire which is wrapped around a metallic core. An electric current is passed through the wire which then creates a magnetic field. **Solenoids** are important in physics because they can be used to create very controlled magnetic fields, resulting in electromagnets. More specifically they can be used to create uniform magnetic fields in a given volume which can then be used for experiments. Below is a wired curled into a loop with its current traveling in a clockwise (to the right) direction: If we were to turn the loop of wire so that the loop went west to east, the wires traveled into and out of the page, added field lines and a cross-section was cut, we would arrive at this: To examine a solenoid mathematically, some variables need to be added. For the below picture, we use **//h//** as the units of length of the solenoid and **//n//** as the number of turns (complete circle in the loop) per that unit of length. Below is a picture of those added variables: In order to measure the magnitude of a solenoid, we need to derive an equation. We start with Ampere’s Law: The enclosed current would be the current of a loop **//i//** multiplied by our variables **//h//** and **//n://** We then substitute and integrate with respect to the length to arrive at: For an ideal solenoid the **//h//** variable will disappear, leaving the equation for the magnitude of the magnetic force of an ideal solenoid: Here is a video of the magentic field caused by a loop of wires conducting electrons: media type="youtube" key="G9Glw3BUTAQ" height="344" width="425" align="center" =Toroids= By bending a solenoid into a loop we create toroidal inductors. These inductors are used in transformers and high-frequency coils. This is what a toroidal coil looks like: [[image:toroid1.bmp align="center"]To find the magnitude of a toroidal loop, we use our equation for solenoids and modify it a bit. Looking at the picture below we see a loop of wires with current **//i//** and shaped in a circle of radius **//r//**. The magnetic force is pushing in a clockwise manner with magnitude **//B.//** [[image:toroid2.bmp align="center"]]
 * Ampere’s Law** is used to measure the magnetic field in a two dimensional plane (closed loop) perpendicular to a current-carrying element. **Ampere’s Law** states that for that closed loop path, the integral of **//B//** equals the permeability constant (µ0) times the current of the element.

Once again we use Ampere’s Law to measure the magnetic force in a closed loop surrounding a wire. We make our closed loop such that, in the picture above, it includes the inner wires but excludes the outer ones (even though they are attached). This loop draws the orange circle. We then use **//N//** to count the number of wires included in our loop (to get a total current for Ampere’s Law). This gives us: Which can be simplified to: Toroidal field video: media type="youtube" key="edqGNOrW1GM" height="344" width="425"