Examples of B-field in the following topics:
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- $E = \frac{1}{2} LI^2 = \frac{1}{2} \frac{\mu N^2 A}{L} \frac{B^2 L^2}{\mu^2 N^2} = \frac{B^2}{2\mu}~AL$.
- (We used the relation $L = \frac{\mu N^2 A}{l}$ and $B = \mu NI/L$. )
- Therefore, the energy density $u_B = energy / volume$ of a magnetic field B is written as $u_B = \frac{B^2}{2\mu}$.
- Magnetic field created by a solenoid (cross-sectional view) described using field lines.
- Energy is "stored" in the magnetic field.
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- Magnetic field stores energy.
- The energy density is given as $u = \frac{\mathbf{B}\cdot\mathbf{B}}{2\mu}$.
- Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field.
- In general, the incremental amount of work per unit volume δW needed to cause a small change of magnetic field δB is:
- Once the relationship between H and B is known this equation is used to determine the work needed to reach a given magnetic state.
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- The magnetic force on a charged particle q moving in a magnetic field B with a velocity v (at angle θ to B) is $F=qvBsin(\theta )$.
- The magnitude of the magnetic force $F$ on a charge $q$ moving at a speed $v$ in a magnetic field of strength $B$ is given by:
- This formula is used to define the magnetic strength $B$ in terms of the force on a charged particle moving in a magnetic field.
- Magnetic fields exert forces on moving charges.
- The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B.
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- A rod is moved at a speed v along a pair of conducting rails separated by a distance ℓ in a uniform magnetic field B.
- Now Δ=Δ(BA)=BΔA, since B is uniform.
- $EMF = \frac{B\Delta A}{\Delta t} = B \frac{l \Delta x}{\Delta t} = Blv$.
- The magnetic field B is into the page, perpendicular to the moving rod and rails and, hence, to the area enclosed by them.
- (b) Lenz's law gives the directions of the induced field and current, and the polarity of the induced emf.
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- Magnetic field lines are useful for visually representing the strength and direction of the magnetic field.
- The magnetic field is traditionally called the B-field.
- A small compass placed in these fields will align itself parallel to the field line at its location, with its north pole pointing in the direction of B.
- (B) A long and straight wire creates a field with magnetic field lines forming circular loops.
- (A) If small compasses are used to map the magnetic field around a bar magnet, they will point in the directions shown: away from the north pole of the magnet, toward the south pole of the magnet (recall that Earth's north magnetic pole is really a south pole in terms of definitions of poles on a bar magnet. ) (B) Connecting the arrows gives continuous magnetic field lines.
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- Current running through a wire will produce both an electric field and a magnetic field.
- For a closed curve of length C, magnetic field (B) is related to current (IC) as in Ampere's Law, stated mathematically as:
- For a short, straight length of conductor (typically a wire) this law generally calculates partial magnetic field (dB) as a function of current for an infinitesimally small segment of wire (dl) at a point r distance away from the conductor:
- $d {\bf B}=\frac {\mu_0}{4 \pi} \frac {Id{\bf l} \times {\bf r}}{r^3}$.
- ${\bf B}=\frac {\mu_0}{4 \pi} \oint_C {\frac {Id{\bf l} \times {\bf \hat{r}}}{r^2}}$.
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- where B is the magnetic field vector, v is the velocity of the particle and θ is the angle between the magnetic field and the particle velocity.
- If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force:
- The electric field surrounding three different point charges: (a) A positive charge; (b) a negative charge of equal magnitude; (c) a larger negative charge.
- The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B and follows right hand rule–1 (RHR-1) as shown.
- The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B.
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- Faraday's law of induction states that changing magnetic field produces an electric field: $\varepsilon = -\frac{\partial \Phi_B}{\partial t}$.
- In a nutshell, the law states that changing magnetic field $(\frac{d \Phi_B}{dt})$ produces an electric field $(\varepsilon)$, Faraday's law of induction is expressed as $\varepsilon = -\frac{\partial \Phi_B}{\partial t}$, where $\varepsilon$ is induced EMF and $\Phi_B$ is magnetic flux.
- The magnetic flux is $\Phi_B = \int_S \vec B \cdot d \vec A$, where $\vec A$ is a vector area over a closed surface S.
- Also, note that in the right hand side$\frac{\partial}{\partial t} (\int \vec B \cdot d\vec A) = \int \frac{\partial \vec B}{\partial t} \cdot d\vec A$ .
- Describe the relationship between the changing magnetic field and an electric field
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- A uniform field is that in which the electric field is constant throughout.
- Equations involving non-uniform electric fields require use of differential calculus.
- For the case of a positive charge q to be moved from a point A with a certain potential (V1) to a point B with another potential (V2), that equation is:
- In uniform fields it is also simple to relate ∆V to field strength and distance (d) between points A and B:
- In this image, Work (W), field strength (E), and potential difference (∆V) are defined for points A and B within the constructs of a uniform potential field between the positive and negative plates.
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- As vector fields, electric fields obey the superposition principle.
- For example, if forces A and B are constant and simultaneously act upon an object, illustrated as O in , the resultant force will be the sum of forces A and B.
- Vector addition is commutative, so whether adding A to B or B to A makes no difference on the resultant vector; this is also the case for subtraction of vectors.
- Electric fields are continuous fields of vectors, so at a given point, one can find the forces that several fields will apply to a test charge and add them to find the resultant.
- Forces a and b act upon an object at point O.