Conversion of Electromagnetic equations from CGS to MKS units

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var('l, W, w, S, Phi, r, x, y, z, v, alpha, beta, gamma, delta, a, b, c, d, f, F_, m_,M_')
var('C, L, R, mu, epsilon, F, m, M, A, B, D, V, E, Sigma, Pow, P_')
var('sigma, I, rho, q, Q, p, P, H, D, pi, J, c, mu_0, epsilon_0')
eqCGS = H - D/c == 4*pi*J/c  
### Put your equation on the previous line ###
eq = eqCGS * (1/2)*mu_0^(-1/2)*pi^(-1/2)  
### Put your multiplier on previous line ###
assume(c>0,mu_0>0,epsilon_0>0)
eq = eq.subs(C==(4*pi)^(-1)*epsilon_0^(-1)*C)
eq = eq.subs(L==(4*pi)*epsilon_0*L)
eq = eq.subs(R==(4*pi)*epsilon_0*R)
eq = eq.subs(mu==mu_0^(-1)*mu)
eq = eq.subs(epsilon==epsilon_0^(-1)*epsilon)
eq = eq.subs(F==(4*pi)^(1/2)*mu_0^(-1/2)*F)
eq = eq.subs(m==(4*pi)^(-1/2)*mu_0^(1/2)*m)
eq = eq.subs(M==(4*pi)^(-1/2)*mu_0^(1/2)*M)
eq = eq.subs(A==(4*pi)^(1/2)*mu_0^(-1/2)*A)
eq = eq.subs(B==(4*pi)^(1/2)*mu_0^(-1/2)*B)
eq = eq.subs(V==(4*pi)^(1/2)*epsilon_0^(1/2)*V)
eq = eq.subs(Phi==(4*pi)^(1/2)*epsilon_0^(1/2)*Phi)
eq = eq.subs(E==(4*pi)^(1/2)*epsilon_0^(1/2)*E)
eq = eq.subs(Sigma==(4*pi)^(-1/2)*epsilon_0^(-1/2)*Sigma)
eq = eq.subs(sigma==(4*pi)^(-1)*epsilon_0^(-1)*sigma)
eq = eq.subs(I==(4*pi)^(-1/2)*epsilon_0^(-1/2)*I)
eq = eq.subs(rho==(4*pi)^(-1/2)*epsilon_0^(-1/2)*rho)
eq = eq.subs(P==(4*pi)^(-1/2)*epsilon_0^(-1/2)*P)
eq = eq.subs(p==(4*pi)^(-1/2)*epsilon_0^(-1/2)*p)
eq = eq.subs(q==(4*pi)^(-1/2)*epsilon_0^(-1/2)*q)
eq = eq.subs(Q==(4*pi)^(-1/2)*epsilon_0^(-1/2)*Q)
eq = eq.subs(H==2*pi^(1/2)*mu_0^(1/2)*H)
eq = eq.subs(D==2*pi^(1/2)*epsilon_0^(-1/2)*D)
eq = eq.subs(J==(1/2)*pi^(-1/2)*epsilon_0^(-1/2)*J)
eq = eq.subs(c==mu_0^(-1/2)*epsilon_0^(-1/2))
eq = eq.expand()
eq = eq.canonicalize_radical()
eq = eq.simplify_full()
eq = eq.expand()
show('Original (CGS) version')
show(eqCGS)
show('MKS version')
show(eq)

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Messages

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var('l, W, w, S, Phi, r, x, y, z, v, alpha, beta, gamma, delta, a, b, c, d, f, F_, m_,M_')
var('C, L, R, mu, epsilon, F, m, M, A, B, D, V, E, Sigma, Pow, P_')
var('sigma, I, rho, q, Q, p, P, H, D, pi, J, c, mu_0, epsilon_0')
eqMKS = -D+H==J  
### Put your equation on the previous line ###
eq = eqMKS *2 * mu_0^(1/2)*pi^(1/2)  
### Put your constant multiplier on the previous line ###
assume(c>0,mu_0>0,epsilon_0>0)
eq = eq.subs(C==(4*pi)*epsilon_0*C)
eq = eq.subs(L==(4*pi)^(-1)*epsilon_0^(-1)*L)
eq = eq.subs(R==(4*pi)^(-1)*epsilon_0^(-1)*R)
eq = eq.subs(mu==mu_0*mu)
eq = eq.subs(epsilon==epsilon_0*epsilon)
eq = eq.subs(F==(4*pi)^(-1/2)*mu_0^(1/2)*F)
eq = eq.subs(m==(4*pi)^(1/2)*mu_0^(-1/2)*m)
eq = eq.subs(M==(4*pi)^(1/2)*mu_0^(-1/2)*M)
eq = eq.subs(A==(4*pi)^(-1/2)*mu_0^(1/2)*A)
eq = eq.subs(B==(4*pi)^(-1/2)*mu_0^(1/2)*B)
eq = eq.subs(V==(4*pi)^(-1/2)*epsilon_0^(-1/2)*V)
eq = eq.subs(Phi==(4*pi)^(-1/2)*epsilon_0^(-1/2)*Phi)
eq = eq.subs(E==(4*pi)^(-1/2)*epsilon_0^(-1/2)*E)
eq = eq.subs(Sigma==(4*pi)^(1/2)*epsilon_0^(1/2)*Sigma)
eq = eq.subs(sigma==(4*pi)*epsilon_0*sigma)
eq = eq.subs(I==(4*pi)^(1/2)*epsilon_0^(1/2)*I)
eq = eq.subs(rho==(4*pi)^(1/2)*epsilon_0^(1/2)*rho)
eq = eq.subs(P==(4*pi)^(1/2)*epsilon_0^(1/2)*P)
eq = eq.subs(p==(4*pi)^(1/2)*epsilon_0^(1/2)*p)
eq = eq.subs(q==(4*pi)^(1/2)*epsilon_0^(1/2)*q)
eq = eq.subs(Q==(4*pi)^(1/2)*epsilon_0^(1/2)*Q)
eq = eq.subs(H==2^(-1)*pi^(-1/2)*mu_0^(-1/2)*H)
eq = eq.subs(D==2^(-1)*pi^(-1/2)*epsilon_0^(1/2)*D)
eq = eq.subs(J==2*pi^(1/2)*epsilon_0^(1/2)*J)
eq = eq.subs(epsilon_0==mu_0^(-1)*c^(-2))
eq = eq.expand()
eq = eq.canonicalize_radical()
eq = eq.simplify_full()
eq = eq.expand()
show('Original (MKS) version')
show(eqMKS)
show('CGS version')
show(eq)

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Messages

Tips and instructions for units conversion

Using the constant multiplier

As written	Sage Cell (line 4)
$\nabla \times \vec{B} = \frac{1}{c}\frac{\partial \vec{E}}{\partial t}+\frac{4\pi}{c}\vec{J}$	`eqCGS = B == c^(-1)E + 4pic^(-1)J`
$\vec{D}_1=\vec{E}_1+\vec{P}_1$	`eqCGS = D == E + P`
$\vec{F}=m\vec{a}$	`eqCGS = f == m_ * a`
$\frac{1}{\sqrt{4\mu_0 \pi}}$	`(1/2)mu_0^(-1/2)pi^(-1/2)`
$\vec{F}=q(\vec{E}+\frac{\vec{v}}{c}\times\vec{B})$	`eqCGS= f == q(E + vc^(-1)*B)`

The code enables you to multiply the entire equation by an overall factor. Here's an example to show how it works. In the "CGS to MKS" code box, on line 4 enter eqCGS = H-D/c==4*pi*J/c.
On line 6, enter eq = eqCGS * 1 (corresponding to a multiplier of 1), and press Evaluate.
In the answer box, you will see:

$-2D\sqrt{\mu_0} \sqrt{\pi} + 2H\sqrt{\mu_0} \sqrt{\pi} = 2J\sqrt{\mu_0} \sqrt{\pi}$ .
Now change the overall multiplicative factor

$1/\sqrt{4 \mu_0 \pi}$ on line 6: eq = eqCGS * (1/2)*mu_0^(-1/2)*pi^(-1/2), and press Evaluate.
You should now see the simpler equation:

$-D+H=J$

Electromagnetic Variables

Quantity	Gaussian	SI	Sage Cell
Mass	$m$	$m$	`m_,M_`
Lenth	$l$	$l$	`l`
Time	$t$	$t$	`t`
Force	$\vec{F}$	$\vec{F}$	`f, F_`
Energy	$W$	$W$	`W`
Energy density	$w$	$w$	`w`
Power	$P$	$P$	`Pow, P_`
Power flow density	$\vec{S}$	$\vec{S}$	`S`
Charge	$q$	$\frac{q}{\sqrt{4 \pi \epsilon_0}}$	`q`
Surface charge density	$\sigma, \Sigma$	$\frac{(\sigma, \Sigma)}{\sqrt{4 \pi \epsilon_0}}$	`Sigma`
Charge density	$\rho$	$\frac{\rho}{\sqrt{4 \pi \epsilon_0}}$	`rho`
Current	$I$	$\frac{I}{\sqrt{4 \pi \epsilon_0}}$	`I`
Current density	$\vec{J}$	$\frac{\vec{J}}{\sqrt{4 \pi \epsilon_0}}$	`J`
Polarization	$\vec{P}$	$\frac{\vec{P}}{\sqrt{4 \pi \epsilon_0}}$	`P`
Electric diple moment	$\vec{p}$	$\frac{\vec{p}}{\sqrt{4 \pi \epsilon_0}}$	`p`
Electric field	$\vec{E}$	$\sqrt{4 \pi \epsilon_0}\vec{E}$	`E`
Potential (Emf, Voltage)	$\Phi, V$	$\sqrt{4 \pi \epsilon_0}(\Phi, V)$	`Phi, V`
D-field	$\vec{D}$	$\sqrt{\frac{4 \pi}{\epsilon_0}}\vec{D}$	`D`
Magnetic field	$\vec{B}$	$\sqrt{\frac{4 \pi}{\mu_0}}\vec{B}$	`B`
Vector potential	$\vec{A}$	$\sqrt{\frac{4 \pi}{\mu_0}}\vec{A}$	`A`
H-field	$\vec{H}$	$\sqrt{4 \pi \mu_0}\vec{H}$	`H`
Magnetization	$\vec{M}$	$\sqrt{\frac{\mu_0}{4 \pi}}$	`M`
Magnetic moment	$\vec{m}$	$\sqrt{\frac{\mu_0}{4 \pi}}\vec{m}$	`m`
Magnetic Flux	$F_{ij}$	$\sqrt{\frac{4 \pi}{\mu_0}}F_{ij}$	`F`
Conductivity	$\sigma$	$\frac{\sigma}{4 \pi \epsilon_0}$	`sigma`
Dielectric constant	$\epsilon$	$\frac{\epsilon}{\epsilon_0}$	`epsilon`
Magnetic permeability	$\mu$	$\frac{\mu}{\mu_0}$	`mu`
Resistance	$R$	$4 \pi \epsilon_0 R$	`R`
Inductance	$L$	$4 \pi \epsilon_0 L$	`L`
Capacitance	$C$	$\frac{C}{4 \pi \epsilon_0}$	`C`

Conversion of electromagnetic equations: CGS to MKS units

MKS to CGS

Tips and instructions for units conversion

Examples

Using the constant multiplier

Electromagnetic Variables